n WW ,. , x .._‘ aowuhérat ‘ . 11%qu z». .‘Euww a»... . in ”hfiwfit A V “fin? r . zuhu. 1.3"”me .. fiifiz} ., a a 5.3th , x: u. .5 , ‘ a... “Jaw )mafxwmuwxmnw, «.1 ‘ , , . ‘ 3w... . .5. «Am? . , , £3”. . h? ... . , ‘ by. mwmmwtm WWW E 13.15.. . . plums}. , Lu: . 92%., «#25. 4. 3w... nr. 3 fia‘fiafixmswmw . t “NV .8 .. was» «um»... 39.. “.3: . y I "5.55%: ‘ . L... . 7!. .n $3512.? 2 u w: .6\: :93}... ‘4‘“ O. \ . up... mzmfli I" {Y K‘ i. ‘ u vi z... ‘ . .1 ’- , .1? . . . 44. ....afl..m .3... I! 1 HUI." p, /l fllhhi . LIBRARY Michigan State University This is to certify that the dissertation entitled Relative bounded cohomology and Relative £1 homology presented by HeeSook Park has been accepted towards fulfillment of the requirements for Ph . D. degree in Mathema I .| 95 MV (A VVO\/ Major professor Date QF/DB/M MS U is an Affirmative Action/Equal Opportunity Institution 0-12771 PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINE return on or beforedate due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 11m gamma.“ Relative bounded cohomology and Relative 61 homology By HeeSook Park DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 2001 ABSTRACT Relative bounded cohomology and Relative 81 homology By HeeSook Park In his renowned paper ‘Volume and Bounded Cohomology’, M. Gromov developed the theory of bounded cohomology and [1-homology of topological spaces. His theory applies both to the absolute case and the relative one. While the theory of absolute bounded cohomology is fairly well understood algebraically from works of R. Brooks and N. Ivanov, this is not so for the relative case. The goal of this paper is to provide the principal algebraic foundations to the theory of bounded cohomology and 61 homology in the relative case. Moreover, we give the proofs of Gromov’s Equivalence theorem and Relative mapping theorem for both relative bounded cohomology and relative 31 homology, which Gromov states in his paper without proofs. We also define locally finite 31 homology of spaces in terms of relative 61 homology of spaces and prove the Gromov’s Vanishing-Finiteness theorem for locally finite 81 homology. To my parents EunSoon Chung and DongGeun Park iii ACKNOWLEDGMENTS I would like to express my gratitude from my heart to my advisor, Professor Nikolai V. Ivanov, for his excellent advice, encouragement, and great patience. I also would like to give my sincere thanks to my thesis committee, Professor Selman Akbulut, Professor Jonathan I. Hall, Professor John McCarthy, and Professor Jon Wolfson, for their support, valuable time and advice. I am very thankful to my dear friend Joan Stamm who has encouraged me and given to me many valuable comments. Finally I am deeply grateful to my family, my parents and my brothers and sisters, for their loving and constant help and support during all those years. iv TABLE OF CONTENTS Introduction 1 Absolute bounded cohomology groups 6 7 1.1 ' Bounded cohomology of groups ..................... 1.1.1 Bounded G-modules ....................... 1.1.2 Relatively injective G-modules .................. 1. 1.3 Resolutions ............................ 1.1.4 Bounded cohomology of groups ................. 1.2 Bounded cohomology of topological spaces ............... 1.3 Amenable groups and amenable coverings ............... 1.3.1 Amenable groups ......................... 1.3.2 Amenable covering ........................ 1.4 Main properties of absolute bounded cohomology ........... Relative bounded cohomology of groups Relative bounded cohomology of spaces 61 homology of groups Relative 81 homology of groups Relative 61 homology of spaces Locally finite £1 homology of spaces BIBLIOGRAPHY 12 13 13 14 14 17 41 52 64 78 85 101 Introduction The absolute bounded cohomology was first defined for discrete groups. It appeared in a version due to P. Trauber of a theorem of Hirsch and Thurston to the effect that the bounded cohomology of an amenable group is zero. Afterwards, M. Gromov [7] defined the bounded cohomology of topological spaces and proved a number of profound theorems about it. Moreover, Gromov [7] applied the theory of bounded cohomology to Riemannian geometry, thus demonstrating the importance of this theory. The proofs in [7] are based on a specific technique developed by Gromov, which he called the theory of simplicial multicomplexes, rather than on standard ideas of algebraic topology. The first step in understanding the theory of bounded cohomology from the point of view of homological algebra was made by R. Brooks [3]. He based his approach on the ideas of the relative homological algebra. However, his approach was incomplete in at least two respects: it did not let one construct the natural norm which one has on bounded cohomology groups precisely and it used Gromov’s fundamental theorem about the bounded cohomology of simply connected spaces. In [8] N. Ivanov improved Brook’s approach using a suitable version of relative homological algebra, modified so that they take into account a natural seminorm in the bounded cohomology. He also proved the Gromov’s vanishing theorem for the bounded cohomology of simply connected spaces by using the results of Bold and Thom, and an analogue of Leray’s theorem about coverings in the theory of bounded cohomology by using the theory of sheaves. The present paper extends the theories of both the absolute bounded cohomology and absolute 81 homology to the relative ones using the ideas of relative homological algebra. While Gromov defined both relative bounded cohomology and relative 81 homology only for a pair of spaces X and Y C X as similar to the ordinary relative cohomology and homology, we define them for continuous map of spaces which also induce the standard cases. For a topological space X, we denote by Sn(X) the set of n-dimensional singular simplices in X. The 81 homology of a topological space X, denoted by Hf‘ (X), is defined as the homology of the complex Cf,‘ (X) of £1 chains c = 22173-0,- with the £1 norm ||c||1 = 2:1 |r,-| < 00, where r,- E R and a,- E 3,,(X). Thus C,’,1(X) is the norm completion of the ordinary chain complex Cn(X) and so it is a Banach space. In [7] Gromov used this 131 norm mainly for defining the simplicial volume of open manifolds. By taking the dual Banach space of Cf,‘ (X), we obtain a cochain complex Hom(Cf,1(X),R) and its cohomology is called bounded cohomology of X and is denoted by H"(X). It is more convenient, in some respect, to deal with the bounded cohomology with the following independent description, which we will use, rather than with the dual space of (31 chain group. We define H"(X) as the cohomology of the complex B‘(X) = {B"(X),0n}, where B"(X) is the space of bounded real valued functions on 3,,(X) (see Section 1.2). We define the norm of f E B"(X) by setting ”f” = sup{ |f(0)| | a E Sn(X)} which turns it into a Banach space. Thus on H"(X) there is a natural seminorm Hall = inf H f H, where f runs over all bounded cocycles representing a E H"(X). There is a group-theoretic analogue of bounded cohomology, which is discussed in detail in [8]. Here we briefly introduce the bounded cohomology of groups by using the standard resolution. For a discrete group G, let B(G'") = {f: G" —> R I ||f|| < oo }, where ||f|| = sup{|f(gl,--- ,gn)| | (g1,~- ,gn) E 0"}. Then B(G’") is a bounded G-module by which we mean B(G") is a real Banach space together with the G-action g-f(gl, - -- ,9”) = f(gl, - -- ,gng) such that ||gf|| _<_ Hf”. Then there is aG-resolution of the trivial G-module R 11—1 0—>R—>B(G)i%B(G2)-d—‘>B(G3)ii>m which is called the standard G-resolution, where the boundary operators are defined by the formulas d_1(c)(g) = c and for every n 2 0 dfl(f)(gli ' ' ' agn+2) : (—1)n+1f(92v ' ' ' ign+2) n+1 + Z(—1)n+1_if(91a”' agigi+1a"' ,9n+2)- i=1 Also we let B(G’")G = {f E B(G") | g-f = f for all g E G}. Then we have acomplex 0 —> B(G)G ——> B(G2)G ——> 3(03)G —> The cohomology of this complex is called the bounded cohomology of G and is denoted by H*(G). An important feature of the theory is that the bounded cohomology of a topolog- ical space and its fundamental group coincide. That makes it possible to study them simultaneously from two View points: group theory and topology. Let cp: Y —> X be a continuous map of spaces. We consider the mapping cone B"(X) GB B"‘1(Y) and its boundary operator defined by dn(unavn—l) : (anuna —)\nun _ 6:1_1vn—l)i where a. and 8; are the boundary operators on B*(X) and B*(Y) respectively and X': B‘(X) —> B*(Y) is a cochain map induced by (,0. Then {B"(X) Q Bn’1(Y), dn} is a complex. We call the n-th cohomology of this complex the n—th relative bounded cohomology of X modulo Y and denote it by HWY 3—) X). We define the norm ll - || 3 on B his (I) semi exam ping 0' ‘ OIL) lSLI] I {in on B"(X) $3"‘1(Y) by setting ||(u,,,v,,_1)|| = max{||u,,||, Hun-1H}- This norm induces a seminorm || - II on H‘(Y 3+ X). Similarly, for a homomorphism of groups 4p: A —> G we define H*(A 2+ G) whose seminorm depends on the choice of a pair of G- and A-resolutions. As an important example, the group H*(A 3) G) is defined as the cohomology of the complex of map- ping cone {B(G"+1)G $ B(A")A} induced from the standard 0- and A-resolutions. The seminorm on which is in fact the canonical one, i.e., the infimum of seminorms which arise from all pairs of G- and A-resolutions which define IAI‘(A 3+ G). If we consider a pair of spaces X and Y C X and an inclusion map (,0: Y <—+ X, then there is an exact sequence n 0 —+ ker(p") 9—) B"(X) 1+ B"(Y) —+ 0, where p" is the cochain map induced by 4,9. It is obvious that {ker(p*)} is a complex. The n-th cohomology of the complex {ker(p*)} is the relative bounded cohomology group, by Gromov’s definition. It is denoted by H"(X, Y). In fact, there is a canonical isomorphism of vector spaces H"(fl): I?”(X,Y) —+ fi"(Y 3) X) which carries the seminorm on H"(X, Y) to a norm equivalent to the seminorm on HWY 1’3 X). In fact, we have an explicit estimate as the following: 1 n+2 lllflll s IIH"(fi)[flll s lllflll for [f] e FI" ker(p") :—> B(G"+1)G ”—3 B(A"+1)" —> 0. 4 Thus, for a pair of spaces X and Y C X, we can define H Wle , 1r1Y) only when the natural inclusion map Y ‘——> X induces an injective homomorphism 7r1Y ——> 1r1X. In this case, there is an isomorphism fin(X, Y) —) ITW‘n'lX, 771Y) which carries the seminorm on HWX, Y) to a norm equivalent to the seminorm on H"(1r1X,1r1Y). The difficulty arises from the fact that the induced homomorphism 1r1Y —> «1X is not injective in general. From our definition, for a continuous map 4p: Y —> X and the induced homomor- phism 90...: 7r1Y —-> 7r1X, we shall construct the cochain map B"(X) EB B"‘1(Y) —> B<n+1>mX EB B((W1Y)")"‘Y and we shall see that the group H WY 3) X) is canonically isomorphic with the group HW‘NIY i’1) 1r1X) and this isomorphism carries the seminorm on H WY 3) X) to the canonical seminorm on H"(1r1Y 31) 1r1X). So the relative bounded cohomology of spaces also coincides with the relative bounded cohomology of their fundamental groups. Thus it appears that our definition of relative bounded cohomology is more natural. Amenable groups, whose definition is recalled in Section 1.3, play a special role in the theory of bounded cohomology. One of the important facts is that the bounded cohomology of an amenable group is zero. We shall see that HWY 3+ X) is isomet- rically isomorphic with PIWX) if the group 1r1Y is amenable. As similar to the relative bounded cohomology, we also define the relative 61 homology groups Hfl (Y 3) X) and Hf‘ (A 3) G) for a continuous map (,0: Y -> X of spaces and for a homomorphism (,0: A —> G of groups respectively by using mapping cones. Then we see that the relative 61 homology of spaces also coincides with the relative 61 homology of their fundamental groups. Now we describe the content of the paper. In Chapter 1, we review the basic def- initions and results of the theory of absolute bounded cohomology following Ivanov’s paper [8], which is the main source of the ideas of the present paper. In Chapter 2, we construct the theory of the relative bounded cohomology of discrete groups following the ideas of relative homological algebra. For a group ho- momorphism (p: A —> G, we define the relative bounded cohomology of G modulo A and denote it by fi*(A 3) G) [Definition 2.3]. We also define the norms || - ”(w) on I?*(A 3+ G) for every w 6 [0,00] and prove that the canonical seminorm coincides with the seminorm induced by the standard G— and A-resolutions [Corollary 2.8]. For an amenable group A, we prove that the groups fi*(G) and 3*(A i’1) G) are isomet- rically isomorphic for the norm || - || [Theorem 2.14] and that the norms [I - ||(w) on fi*(A 3) G) are equal for every w 6 [0,00] [Theorem 2.15]. Also, for a subgroup A of G and the natural inclusion map (p: A ¢—) G, we prove that the group fi*(G, A) is isomorphic to I?*(A 3+ G) and the isomorphism carries the seminorm on fi*(G, A) to the norm equivalent to the canonical seminorm on ETA 3+ G) [Theorem 2.19]. It is not known if they are actually equal. In Chapter 3, we define the relative bounded cohomology of a space X modulo Y of a continuous map Y 3) X and denote it by fi*(Y 3) X) [Definition 3.2]. Also we define the norms || - ||(w) on §‘(Y 3+ X). The main result of this section is that I?‘(Y 3) X) and I?“ (1r1Y fl) 1r1X) are isometrically isomorphic [Theorem 3.3]. We also prove in Corollary 3.4 and Theorem 3.5 respectively Equivalence theorem and Relative mapping theorem which are stated in the Gromov’s paper [7]. In Chapter 4, we construct a theory of 81 homology of discrete groups. We denote the [31 homology of a group G by Hf1(G). The main results of this section are that the canonical seminorm in H? (G) coincides with the seminorm induced by the bar resolution [Theorem 4.2] and that H51 (G) is zero if G is amenable [Theorem 4.10]. In Chapter 5, we define the relative 81 homology of a group G modulo A for a group homomorphism A f) G and denote it by Hf1(A 3+ G)[Definition 5.3]. We prove that the groups Hf1(G) and Hfl(A 3) G) are isometrically isomorphic for the norm l] - ||1(w) for an amenable group A [Corollary 5.10] and that the norms || - ||1(w) on Hf1(A 31> G) are equal for every w 6 [0,00] for an amenable group A [Theorem 5.8]. For a subgroup A of G, we also define Hf‘(G,A) following the usual approach to the relative homology of groups. Then we prove that the groups Hfl (G, A) and Hf‘ (A 3+ G) for the natural inclusion map (p: A ‘—> G are isomorphic and this isomorphism carries the seminorm on Hfl (G, A) to a norm equivalent to the canonical seminorm on H ,9 (A 3) G) [Theorem 5.12]. In Chapter 6, we define the relative 21 homology of a space X modulo Y of a continuous map cp: Y —> X and denote it by Hf‘ (Y 3) X) [Definition 6.3]. The main result of this section is that the groups Hf1(Y 3) X) and H51 (7r1Y fl) 1r1X) are isometrically isomorphic [Theorem 6.4]. In Chapter 7, we define the locally finite 31 homology group of a space X as the inverse limit of an inverse system of relative (31 homology groups H51 ((X - K j) H X) for every compact subspaces K,- C X and the inclusion maps X — K,- H X. We denote it by Hf°(X) [Definition 7.1]. We also define the norms II ' “a and I] - ”(0.2) on Hf°(X) [Definition 7.2]. The group Hf°(X) is an analogue of the group FAX) in [7] which Gromov defined with the locally finite singular Kl-chains c = 2:, 73-0,- such that each compact subset K of X intersects only finitely many (images of) simplices 0,. On FAX) there are the norms II - H1 and [I - ”1(0) for every 9 6 [0,00] induced by the norms “CH1 2 2:10,] and ||c||1(9) = IICH1 + fillaclll respectively. In fact, for every 9 6 [0,00], Gromov defined the norm ||h||1(6) of h E F...(X) as the limit limjaoo ||hj||1(6l), where hj 6 H..(X, U,-) is the homomorphic image of h E F.(X) for a sequence of subsets U j C X which are large, i.e., the closure of X -— U,- is compact, and such that only finitely many U j intersects any given compact subset of X. It seems plausible that the groups H f°(X ) and 71-...(X ) are isomorphic and the norms I] - [I3 and || - ||,(w) are equivalent to the norms H - [[1 and II - ”1(0) for w = 0 respectively but we leave it as an open question. We consider an amenable covering of X. A subset Y of X is called amenable if for every path connected component Y’ of Y the image of the inclusion homomorphism 7r1Y’ -—> 1r1X is an amenable group. A covering of a space X is amenable if all its elements are amenable (see Section 1.3.2). We prove Equivalence theorem, which is only stated in the Gromov’s paper [7], to the effect that on Hf°(X) the norm || - I], is equal to the norms || - ||z(w) for every w E [0, 00] if a space X is amenable at infinity, i.e., every large set U C X contains another large amenable subset U’ C U [Theorem 7.5]. Finally, let a space X admit an amenable covering. Then we prove the Gromov’s vanishing theorem in [7] for Hf‘ (X) and for Hf°(X) to the effect that the norm || - ”1 on H51 (X) and the norm || - I], on Hf(X) are equal to zero for n 2 m if every point of X is contained in at most m elements of this covering [Theorem 7.7 and Corollary 7.8]. We also prove the Gromov’s finite theorem for HS°(X) to the effect that the norm || - H, on HS" (X) is finite for n 2 m if there is a large set every point of which is contained in at most m elements of this covering [Theorem 7 .9]. In [7] Gromov proved Vanishing-Finite theorem for the group FJX) using locally finite diffusion operators p*, where ,u is a non-negative real valued function on a group G such that ||u||1 = Z |p(g)| = 1. Then for an 61 function f on X and a group G acting on X, the diffusion operators are defined as (u * 003) = 2960 #(g)f(9“rc) for III E X- CHAPTER 1 Absolute bounded cohomology groups In this chapter, we review the basic definitions and results of the theory of bounded cohomology in [8]. Throughout this chapter G denotes a discrete group. 1.1 Bounded cohomology of groups 1.1.1 Bounded G-modules By a bounded left G module we mean a real Banach space V together with a left action of G on V such that [lg - v” S [[0]] for all g E G and v E V. We define a bounded right G-module analogously. We shall call a bounded left G-module simply G-module. For two G-modules V and W, a bounded linear operator V —> W which commutes with the action of G is called a G-morphism. The simplest and also an important example of G-module is R, considered together with the trivial action of G. Another important example of G-module is B(G") the set of all bounded functions f: G" -—> R, where G" = G x G x x G is considered together with the action: \__‘,.___/ fl g ' f(gla° ° ' ign—lygn) = f(glv' ° ' agn—lagngl' More generally, for any Banach space V, we consider the space B (G, V) of func- tions f: G —> V such that ||f|| = sup{|f(g)| I g E G} < 00. Then B(G,V) is a Banach space with the norm || ~ [I and the action defined by g - f (h) = f (hg) turns it into a G-module. It is clear that the space B (G"+1) is isomorphic with B (G, B (G")), where B (G") is considered simply as a Banach space. 1.1.2 Relatively injective G—modules An injective G-morphism of G-modules t: V —-> W is said to be strongly injective if there exists a bounded linear operator 0: W —> V such that a 02' = id and “all 3 1. We call a G-module U relatively injective if, for any strongly injective G—morphism of G-modules i: V —> W and any G-morphism of G—modules a: V —+ U, there exists a G-morphism 6: W —+ U such that 6 02' 2 oz and ”fill S Hall. For example, for any Banach space V, the G-module B (G, V) is relatively injective. In particular, the G-modules B(G") are relatively injective (see Lemma 3.2.2 in [8]). 1.1 .3 Resolutions By a strong relatively injective G-resolutz'on of a G-module V we mean a sequence of G-modules and G-morphisms of the form Osvflvoflvlikvzflm which is exact as a sequence of vector spaces and satisfies the following two conditions: i) the sequence is provided with a contracting homotopy, i.e., a sequence of bounded linear operators kn: Vn —> Vn_1 such that dn_1 0 kn + kn“ o dn 2 id for every n 2 0 and kg 0 d_1 = id and also I] kn [IS 1; ii) every G—module Vn is relatively injective. 10 We consider a G-resolution of the trivial G-module R of the form d-.1 d0 2 d1 3 d2 0—)R——>B(G)—+B(G)—sB(G)—+m, where the boundary operators are defined by the formulas d_1(c)(g) = c and dn(f)(91,-~ ,gn+2) = (-1)"+1f(92,-~ ,gn+2) n+1 + Z(_1)n+l—if(gla ° ° ' igigi+la ' ' ° ign+2)' i=1 This G-resolution becomes strong relatively injective with the contracting homotopy R+k—B(G)<:—B(Gz)<——~-, k2 where kn(f)(gl,--- ,gn) = “91,": ,9", 1). We call this resolution the standard G- resolution. It will play an important role in the theory of bounded cohomology. 1.1.4 Bounded cohomology of groups For any G-module V, we denote by VG the space of G-z'nvariant elements in V, i.e., VG={v€V|g-v=u forallgEG}. For any strong relatively injective G- resolution of the trivial G-module R 0—>R——>V0—>V1—+V2—>~- the induced sequence 0—>VOG—>VIG—>V2G—+ is a complex and the cohomology of this complex depends only on G. Its n-th co- . homology group is called the n-th bounded cohomology group of G and is denoted by ff"(G). Note that in ff”(G) there is a natural seminorm which induces a natural topological vector space structure on it. Also note that this seminorm depends on the choice of resolution. 11 We define the canonical seminorm on ff“(G) as the infimum of the seminorms which arise from all resolutions. In fact, this infimum is achieved by the standard resolution, i.e., the seminorm on 17*(G), defined by the standard G-resolution, coin- cides with the canonical seminorm (see Theorem 3.6 in [8]). Remark that f1*(G) is a contravariant functor of G. 1.2 Bounded cohomology of topological spaces Let X be a topological space. For every n 2 0, we denote by Sn(X) the set of n- dimensional singular simplices in X and by G"(X) the real n-dimensional singular cochain group, i.e., the set of arbitrary functions 5,,(X) —> R. As is well known, the sequence 0 —> 0°(X) 3+ 0‘00 3 C2(X) 35% is a complex, where d is defined by d, f (a) = 2?:01(—1)‘f(8,-o) and (9,0 is the i-th face of the singular simplex 0'. The cohomology of this complex is H *(X ) the real singular cohomology group of X. Let B"(X) C C"(X) be the space of bounded functions Sn(X) —> R. Its elements are called bounded cochains. There is a natural norm || - I] in the space B"(X) given by I] f I] = sup{] f (0)] | o E Sn(X)} which obviously turns it into a Banach space. It is clear that dnB"(X) C B"+1(X). The cohomology of the complex 0 —+ B°(X) fl Bl(X) —> is called the bounded cohomology of X and is denoted by U*(X ). In fl*(X) there is a natural seminorm I] - I] defined by ||[c]|| = inf M f H for a cohomology class [c] of ff*(G), where the infimum is taken over all bounded cochains f in B"(G) lying in the cohomology class [c]. Remark that the inclusions B"(X) g) 12 G"(X) induce a canonical map I?*(X) —> H *(X ), which in general is neither injective nor surjective. The first basic result of the theory is that the bounded cohomology of a simply connected space is equal to zero (see Theorem 2.4 in [8]). Moreover, ff“(X) depends only on the fundamental group 1r1(X) (see Theorem 4.1 in [8]). 1.3 Amenable groups and amenable coverings Amenable groups play a special role in the theory of bounded cohomology. The tech- nical aspect of this role is, roughly speaking, that for bounded functions on amenable groups one can define their mean value in a natural way. One of the important facts is that the bounded cohomology of an amenable group is zero (see Theorem 3.8.4 in [8])- 1.3. 1 Amenable groups Let S be a set. As is well known, the space B (S) of all bounded functions on S is a Banach space with the norm I] f I] = sup{ | f (23)] I :1: E S }. A linear functional m: B(S) —> R is called a mean if inf{f(:c) | :1: E S} S m(f) S sup{f(:z:) | :L' E S} for all f 6 8(3). Let the group G act on S on the right. Then G acts on B(S) on the left by the formula 9 . f(s) == f(s - g), where g E G, f 6 8(5), and s E S. The mean m on B(S) is called right-invariant if m(g - f) = m(f) for all g E G, f 6 8(5). If there is a right-invariant mean on B (G), then the group G is called amenable. The important facts are that abelian groups and the homomorphic images of amenable groups are amenable. 13 1 .3.2 Amenable covering A connected subset Y of the space X is called amenable if the image of the inclusion homomorphism 1r1(Y) —> 1r1(X) is an amenable group. An arbitrary subset of the space X is called amenable if all its components are amenable. Finally, a covering of the space X is called amenable if all its elements are amenable and in addition it satisfies the following conditions i) and ii): i) the space X, all elements of the covering, and all their finite intersections, are homotopy equivalent to countable cellular spaces; ii) either the covering is open or it is closed and locally finite and the space X is paracompact. For example, if 1r1(Y) is amenable, then Y is amenable. 1.4 Main properties of absolute bounded cohomol- 033’ Now, we state the main results of the theory of absolute bounded cohomology. We refer the proofs to [8]. Theorem 1.1. Let X be a countable cellular space. If X is simply connected, then §"(X) = 0 for alln Z 1. Theorem 1.2. Let A be an amenable normal subgroup of G. Then the map 90*: ff‘(G/A) —> ff“(G), induced by the canonical homomorphism 90: G —> G/A, is an isometric isomorphism, i.e., it preserves the canonical seminorm. Corollary 1.3. IfG is amenable, then ff"(G) = 0 for all n 2 1. Theorem 1.4. Let X be a connected countable cellular space. Then fil"(X) is canon- ically isomorphic with I’l*(1r1X). The seminorm in fi*(X) is carried to the canonical seminorm in if *(7r1X ) through this isomorphism. 14 Proof. For our further use, we sketch the proof of this theorem. Let 7r: X —> X be a universal covering of X. Let I] - || denote the canonical seminorm on if *(1r1X ) and let I] ' H, the seminorm on fi*(X). First, it is shown that the sequence 0 —+ R —> B°(X) —> B1(X) —> 32m) —> (1.4.1) is a strong relatively injective 7r1X-resolution of the trivial 1r1X-module R. Since the map 7r“: B“(X) —> B‘(X) establishes an isometric isomorphism B*(X) ——> B‘(X)"1X and commutes with the boundary operators, the bounded cohomology F1 *(1r1X ) induced from the complex 0 _> BO(x)1r1X __) Bl(x)1r1X __) B2(x)1nX __> B3(x)1r1X _) . . . coincides with ff*(X) as topological vector spaces. Remark that the seminorm on fl*(X) coincides with the seminorm on if ‘(1r1X ) induced by the resolution in (1.4.1). So we have [I . [I g [I - “5. On the other hand, for every n 2 0, it is shown that there is a 1r1X -morphism C": B((1r1X)"+1) —+ B"(X) of the resolutions 0 ——> R ———> B(1r1X) —-—> B((1r1X)2) —+ B((1r1X)3) ——+ l .1 <°l <11 <21 0 ——> R —-> BO(X) —> Bl(X) ——+ B2(X) ——> extending idR and such that ||C"|| g 1 for n 2 0. Namely, let X 0 C X consist of one element from each 1r1X-orbit. For each singular simplex o: A" —> X we set {a} = (g0,--- ,gn), where g,- E 1r1X such that o(v,) E gn_,----ano and v,- is the i-th vertex of the simplex An. Then we define C" by the formula C"(f)(o) = f ({0}) for every f E B((1r1X)"+1). Since ||C"|| g 1, this shows that we have N - ll,, 5 || - ||. Hence the seminorm in fi*(X) is equal to the canonical seminorm in ff“(1r1X). Cl 15 071 (ll mom! Thec Corollary 1.5. The group ff‘(X) is zero if 1r1X is amenable. Theorem 1.6. Let X1 and X2 be connected countable cellular spaces and let f: X1 —> X2 be a continuous map. If the homomorphism f...: 1r1X1 —> 1r1X2 is a surjection with an amenable kernel, then the homomorphism f’“: ff‘(X2) —> if "(X1) is an isometric isomorphism. Theorem 1.7. Let X be a topological space, M be an amenable covering of the space X, N be the nerve of this covering and [N] be the geometric realization of the nerve. Then the canonical map ff”(X) —> H*(X) factors through the map H" (IN]) —> H *(X ) Corollary 1.8. If X admits an amenable covering such that each point of X is con- tained in no more than m elements of the covering, then the canonical homomorphism ff"(X) —> H"(X) vanishes for n 2 m. 16 CHAPTER 2 Relative bounded cohomology of groups Throughout this chapter, G and A denote the discrete groups. Let (p: A —-> G be a group homomorphism. Then any G-module U can be made into an A—module by defining the action of A to be a - u = 90(a) - u for a E A and u E U. We recall that ff‘(G) is a contravariant functor of G. This functoriality can also be described in terms of arbitrary resolutions, as follows: Let 0—>R——>U0—+U1—>--- and 0—>R—>%—H/1—>--- be strong relatively injective G- and A-resolutions of the trivial G- and A-module R, respectively. Then the map idR extends to an A-morphism of resolutions, i.e., to a commutative diagram: O——>R——>U0———>U1——>--- l at .1 All 0————-)R——>V0—-—>V1__>... where Ai(cp(a) - u) = a - Af(u) for a E A and u E U. It follows from the last formula that A‘(U,G) C MA, and hence there is an induced homomorphism go“: ff“(G) —> ff"(A), which depends only on go (see Lemma 3.3.2 in 17 [8]). Also remark that ”99’“ S 1. Definition 2.1. Let 4p: A —> G be a group homomorphism. A strong relatively injec- tive G-resolution of a G-module U a- 0—)U—il U093>U1$U222+ (— <— <— <— k0 k1 k2 k3 and a strong relatively injective A-resolution of an A-module U 8’_ a' 0' o' 0—+U———>1Vo—3>V1—1>V2—3—)~~ (— <— <— <— ‘0 t1 t2 t3 are called an allowable pair of resolutions for (G, A; U) if idu can be extended to an A-morphism of resolutions A": Un —> V, such that A" commutes with the contracting homotopies kn and t,, for all n 2 0. Proposition 2.1. Let (,0: A —> G be a group homomorphism. The standard G- and A- resolutions of the trivial G— and A-module R are an allowable pair of resolutions for (G, A; R). Proof. Recall that the standard G- and A-resolutions o—>REL>B(0):‘9+B(G2)$B(G3)$>M, <— <— +— <— k0 k k2 k3 t-l 0—>R£‘—>B(A) {—— to t B(Az) is B(A3) ill. <— <— ‘2 13 TE" g-I of the trivial G- and A-module R are strong relatively injective. Also recall that the contracting homotopy kn: B(G"+1) —> B(G") is defined by the formula kn(f)(gli"' ign) : f(gli'” 397171) and that G acts on B(G") by 9-f(91.-~ .91.): f(91,--- .9119)- 18 It suffices to show that there is an A-morphism of the standard G-resolution to the standard A-resolution 0 ——> R 11> B(G) is B(Gz) —dl——> B(G3) is l a pot pl p21 0 ———> R is B(A) i B(A2) i B(A3) ——> extending idR and such that p” commutes with the contracting homotopies kn and tn for every n 2 0. We define p" by the formula pn(f)(a1i ' ' ' aan+1) : f((P(0:1), ' ' ‘ 390(an+1)) for f 6 B(G"+1) and (a1,---an+1) E An“. It is easy to check that p" commutes with the boundary operators and has the norm ||p"|| S 1. Note that, for a e A and f e B(G"+1), we have p"(a-f)(a1,--- .an+1) =p”(s0(a) -f)(a1,~~ .an+1) = (90(a) - f)( G be a group homomorphism. Let O—sRaneUl-sm and O—>R—>V0—->V1—>-~ be an allowable pair of resolutions for (G, A; R). For every n 2 0, the mapping cone M"(A 3) G) and the mapping cylinder EM"(A 3) G) of the cochain complexes induced by (,0 are defined as follows: M"(A 3) G) = (15$an EM"(A 2+ G) = V: 63 US 69 VJ}... for every n 2 0 and where V_A = 0. Lemma 2.2. Let (,0: A —+ G be a group homomorphism. Let OsRisvoisvlim and OsRiin13..- be an allowable pair of resolutions for (G,A;R), and let A": Un —> Vn be an A- morphism of resolutions commuting with the contracting homotopies. Then the se- quences 0sM°(Ast) $100130) 3 M2(Af+G) 12s (2.2.1) o s EM°(A 3; G) i EM‘(A 3s G) is EM2(A 1’. G) i) (2.2.2) are complexes, where the boundary operators dn are defined by the formulas dn(u,,,v,,_1) = (anun, —)\"un — 8;,_1v,,_1) on M"(A 3) G) dn(v,,, un,v,,_1) = (agvn, anun, vn -— Anun — 8;,_1vn_1) on EM"(A 31) G). 20 Proof. We check dann = 0 for EM"(A 3) G). dn+1dn(vn, un, vn__1) = dn+1(02’2vna anuna ’Un — Anun - ail—10ml) = ( :,+18,',,vn, 0n+1anum 8:,vn — An+1(3,,u,,) — 8:,(vn — Anun — 8:,_1vn_1)) = (o, 0, 0,12), —- Wlanun — 8:32.. + 89%.. + Btaiflvn—l) = (0, 0, 0). By the same way, it is easy to check dn+1dn = 0 for M ”(A 3) G). El Definition 2.3. The n-th cohomology of the complex in (2.2.1) is called the n-th relative bounded cohomology of G modulo A and is denoted by [TWA 3) G). Also the n—th cohomology of the complex in (2.2.2) is denoted by U"(EM(A 3) G)). We define the norm || - I] on EM"(A f> G) = VnA Q US®VnA_1 by setting “(UmUm‘Un—llll = maxillvnlls llunllv [Iva—1H}, and similarly on M"(A 3) G) by setting [](u,,,v,,_1)|| = max{||u,,||, ||vn_1||}. Remark that these norms define the seminorms || - [I on fi*(A 5 G) and I? *(EM (A 3) G)) respectively. Furthermore, for every w 2 0, we define a norm [l - [[(w) on M"(A 3) G) by putting “(um vn_1)||(w) = maXUlunIL (1+ w)"1||vn—1||}- Observe that all norms || - ”(G)) are equivalent to the norm [I - [I = l] - “(0). Now with this norm on M"(A 3) G) we have the corresponding norm I] - ”(02) on U‘(A 3) G). Finally we define this norm || - ”(10) on I?*(A 31) G) for all (0 in the closed interval [0, 00] by passing to the limits. Proposition 2.3. Let (,0: A —> G be a group homomorphism. Let _ l 8’— al at (i-sni‘svofl’sv1 9s... and OsR——‘>V0—°sV1 —‘>.-. 21 be an allowable pair of resolutions for (G,A; R) and let A": Un —> V,, be an A- morphism commuting with the contracting homotopies. Then the natural projection map p": Elll"(A 3) G) —> US induces an isometric isomorphism A H"(p): F1"(EM(A 3) G)) s H“(G). Proof. We consider the exact sequence OsVHAEBVA csEM"(A =VAQ3UG®VnA A" fso. It is easy to check Vn" ® V,,A_1 is a complex. If (vn, vn_1) is a cocycle of the complex VnA®VnA_1, then we have 0 = d(v,,,vn_1) = (Baumvn — 8:,_1v,,_1) and so vn = 6;,_1’Un_1. Thus (”Un,’Un_1) = (8;,_lv,,_1,v,,_1) : dn_1(’Un_1,0) and thus (Un,’Un_1) IS a coboundary. This shows that the cohomology of the complex VnA ® V41 1 vanishes so that the map H "(p) is an isomorphism. For (vn,u,,,v,,_1) E EM"(A 3) G) = VnA®Uf$VnA_1, we have llpn(vmumvn—1)ll : Hun“ S maxlllvnlla Hun“, Hun—1H} : “(vmumvn—llll- This shows that ||p"|| S 1 and so ||H"(p)|| g 1. On the other hand, we define a mapp U G —> V? 6 U G @ Vn’il by the equation firm) 2 (X‘un, an, 0). Then we have dnpnun = (BLAnunfinunfl) = (A"+13nun,8nun,0) = ”Hanna and so the map p commutes with the boundary operators. It 18 clear that p"p" =id Note that, for every un E US, we have ||5"('un)ll = ||(/\"umum0)|l = max{||)\"unll, llunll,0} S llunll and so ”5"” S 1. This shows that the inverse map (H "(p))‘1 of H"(p) also has the norm II(H"(p))_lll S 1. Hence the isomorphism H"(p) is an isometry. Cl 22 Let (,0: A —+ G be a group homomorphism, and let 6—1 00 a. 6i: 86 0; 0—)R-—>U0—>U1—>---and O—>R—>V0—>V1-—>--- be an allowable pair of resolutions for (G, A; R). Remark that there is an exact sequence of complexes 0 —> M"(A 3) G) 3) EM"(A 15> G) 3) V," s 0, (2.1) where in and p" are natural inclusion and projection maps respectively. This exact sequence in (2.1) induces a long exact sequence —> I?"‘1(A) s I7"(A 3) G) s f1"(G) s EMA) s . (2.2) Recall that the canonical seminorm on ff‘(G) is defined as the infimum of the seminorms which arise from all strong relatively injective G—resolutions of the trivial G—module R. Theorem 2.4. The canonical seminorm on l-l“(G) is induced by the standard G— resolution. Proof. Let 6_ 0—)R—)1 UofliUl—aiiUgéim <— <— +— <— ko k1 k2 k3 be a strong relatively injective G-resolutions of the trivial G-module R. From Theorem 3.6 in [8], it is proved that there is a morphism an from this resolution to the standard resolution 0———->R——> U0 ———> U1———> U2 —-+--- 1 Mal 001 «1 ml 0 ——+ R ——+ B(G) ———+ B(G2) —-> B(G3) —> extending idR and such that ”an” S 1 for every n 2 0, where the morphism an is defined by the formula an(fl(91a ' '° ,gmgnH) = k0(91 ‘ k1(' ' ' (kn—1(gn ' kn(gn+1 ' flll ‘ ' ' )- This finishes the proof. Cl 23 Corollary 2.5. The seminorm on H*(EM(A 3) G)), induced by the standard reso- lutions, coincides with the canonical seminorm on H ’(G). Proof. Remark that the standard G- and A- resolutions define the complex EM"(A X) G) = B(AA+1)A @B(GA+1)G® B(A")A and the cohomology of which is H*(EM (A 3+ G )) Hence it follows from Proposition 2.3 and Theorem 2.4. [:1 Remark that, for a group homomorphism (p: A ——> G, a seminorm on H*(A 3) G) depends on the choice of an allowable pair of resolutions for (G, A; R). Definition 2.4. We define the canonical seminorm || - ”(01) for every w 6 [0,00] on H‘(A 3) G) as the infimum of the seminorms which arise from every allowable pair of resolutions for (G, A; R). As in H*(G) and Lil ‘(EM (A 53+ G)), we shall see the canonical seminorm on PI*(A i) G) is also induced by the standard resolutions. Lemma 2.6. Let (,0: A ——> G be a group homomorphism. Let 6-1 60 31 8:1 86 6; O—sR—>U0-—>U1——>--- and O—sR—AVO—sVI—sm S.— ‘T t.— ‘7 t— A.— 0 1 2 0 l 2 be an allowable pair of resolutions for (G,A;R), and let A": Un —> V" be an A- morphism of resolutions commuting with the contracting homotopies. Then there is a commutative diagram A" U6 VA l l B(Gn+1)G P" 3 B(A"+1)A, where p" is defined by p"f(a1,--- ,an+1) = f(<0(a1), - -- ,so(an+1))- 24 Proof. In the proof of Theorem 2.4, there is a cochain map an: US —> B(G"+1)G defined by the formula 01n(f)(91, ° " agmgn+ll = k0(91'k1("'(kn-1(9n'kn(9n+1'fl)l"'l- Similarly, we define a cochain map 7": VnG —> B (A"+1)A. It is clear that they have the norms Horn” 3 1 and “7,,” S 1. We prove that 'y'nX‘ = pnan. Let f E US and (a1, - -- ,amanH) E An“. Then inA”(f)(a1,--- .an.an+1) =t0(a1 -t1(-~(tn_1(an-tn(an+1-A”(f)))))-~) =t0(a1~t1("°(tn-1(an'tn()\n(90(an+1)'fllll"') =t0(a1.t1(~-(t,,_1(a,,o/\"”1kn((,0(a,,+1)-f))))---) =to(a1-t1(---(tn_1A"‘1(r(an)°kn(R—>Uo—>U1—>--- and 0—)R—>Vo—+V1—>--- 25 be an allowable pair of resolutions for (G,A;R), and let A": Un —> V, be an A- morphism of resolutions commuting with the contracting homotopies. Note that we have two complexes US ® Vn’il and B (G'H'l)G Q B (A")A which are induced from the resolutions above and the standard resolutions respectively. It is enough for us to prove that there is a cochain map 0,: US Q V,,A_1 s B(G"+1)G Q B(A")A such that it has the norm Hflnll _<_ 1. For simplicity, we denote all boundary operators by the same notation d. Recall that, from Lemma 2.6, there is a commutative diagram G A" A Un ———> V" a..[ [7. B(Gn+1)G _L) B(An+l)A in which the maps have the norms Haul] g 1 and “7,,” g 1. We define 5,, by 5,,(un,v,,_1) = (anun, 7,,_1v,,_1). Then we have 5n+1dn(um Uri—1) = fin+l B(G") is a G-morphism and it has the norm ”2"” =1- Lemma 2.9. Let A be an amenable subgroup of G. Then there is a G—morphism 7r": B(G") —+ B((G/A)") such that rrni" = id and ||7r"|| S 1. Proof. From Lemma 3.8.1 in [8], it is proved that there exists a G-morphism 7r: B(G) —-> B(G/A) such that it oi = id and ”it“ S 1. In fact, 71' is defined by the formula 7r(f)(Ag) = mg(f|Ag), where mg is a mean on B(Ag) induced from a right invariant mean on B (A) Also from Corollary 3.8.2 in [8], it is proved that this G- morphism 7r provides a G-morphism 7r": B(G") —> B( (G /A)") such that 7r" 0 i" = id and ||7r"|| S 1 for every n 2 1. Cl Proposition 2.10. Let A be an amenable subgroup of G. Then the sequence 0 s R 1’: B(G/A) gs B((G/A)2) :—_> B((G/A)3) :13 (2.10.1) 27 is a strong relatively injective G -resolution of the trivial G -module R, where the bound- ary operators dn and the contracting homotopy kn are defined as the some ways with the standard resolution. Proof. It is easy to check that the sequence in (2.10.1) is a strong G-resolution. By using the G-morphism 7r": B(G") —) B((G/A)”) in Lemma 2.9 and the fact that B(G") is the relatively injective G-module, it is proved that B((G/A)") is the relatively injective G-module from Lemma 3.8.3 in [8]. [3 Note that the resolution in (2.10.1) induces the complex 0 —> B(G/A)G —> B((G/A)2)G —> B((G/A)3)G ——> (2.3) and the cohomology of which is H*(G). Corollary 2.11. Let A be an amenable subgroup of G. Then the seminorm on H‘(G), induced by the complex in (2.3), coincides with the canonical seminorm. Proof. In H*(G), let I] - [IC denote the canonical seminorm induced by the standard G-resolution and let [I - [I denote the seminorm induced by the complex in (2.3). Recall that there is the canonical G-morphism i": B ((G /A)") —> B (G") such that it has the norm ||i"|] = 1. This shows that H - [[6 S [l - I]. Also, from Lemma 2.9, there is a G-morphism 7r”: B(G") ——> B((G/A)") such that 7r" 0 i" = id and ||7r"'|] S 1. This shows that I] . I] S I] - “C. [3 Note that A /A, the set of cosets of A in A, consists of only one element which we will denote by {A}. Hence B ({A}") consists of all bounded functions on one element {A}" and so it is isomorphic with R. If A is an amenable group, then there is a strong relatively injective A—resolution of the trivial A-module R o s R is B({A}) i B({A}2) is B({A}3) i) B({A}4) is . 28 This resolution induces the complex 0 s B({A})A is B({AmA is B({AW i B({AW is (2.4) and the cohomology of which is H*(A). From definition of the boundary operators, dn(f)({A},--- ,{A}) is the n + 2 h—w—l n+2 alternating sum of f({A}, - -- ,{A}) and so yap—z n+1 id if n is odd, ELI 3 || 0 if n is even. Note that this gives another proof of the bounded cohomology of an amenable group is zero. Corollary 2.12. Let A be an amenable subgroup of G. Then the sequences 0 —> R —> B(G/A) s B((G/A)2) s B((G/A)3) —+ and 0 —+ R —+ B({A}) —> B({A}2) s B({A}3) —-> are an allowable pair of resolutions for (G, A; R). Proof. We define a map p": B((G/A)"+1) —> B({A}"+1) by the formula pn(f)({A},... ,{A}) :f(Ai"' aA) n+1 n+1 It is clear that p" is an A-morphism of resolutions extending idR. Also, as the same way we proved in the standard resolutions, it is easy to check that p" commutes with the contracting homotopies. Cl Proposition 2.13. Let A be an amenable subgroup of G. Let (p: A H G be a natural inclusion map. The seminorm || - ||(w) in H‘(A 3+ G), induced by the complex B((G/A)"+1)G®B({A}")A, coincides with the canonical seminorm for every w 6 [0,00]. 29 Proof. For all (0 Z 0, let II - I|c(w) denote the canonical seminorm in PI*(A 3) G) and let I] - |I(w) denote the seminorm in PI*(A 31> G) induced by the complex B((G/A)"+1)GQB({A}")A. By definition of the canonical seminorm, we have II - I|C(w) S [I - “(01). So, it suffices for us to show that II . ”(01) S [I - I|c(w). From Theorem 2.7, the canonical seminorm in H‘(A 1’1, G) is induced by the complex B (G"+1)G ® B (A")A. Recall that there are canonical chain maps an: B((G/A)n+1)G _) 3(0n+1)G and 711: B({A}n+1)A __) B(An+1)A such that I|a"|| = 1 and ”7"” = 1. Also, from Lemma 2.9, there are chain maps fin: B(Gn+1)G __> B((G/A)n+1)G and An: B(An+l)A _> B({A}n+1)A such that find" = id and Any" 2 id and they have the norms |I7r"|| S 1 and “An” S 1. We consider the following diagram B(gn+1)G _‘Zn__) B(An+l)A ""I [,.. B((G/A>"+1>G P—"> B({A}"+1)A in which q" and p" are defined as the restriction maps so that they have the norms ”an _<_ 1 and “19"” S 1. We prove that this diagram is commutative. Note that A"q“f({A}, ° -- ,{A}) = m(61"f)= m(flA"+1) =7r"f(A,... ,A) =P"7r"f({A}v°° ,{A}), where m is a mean on B(A”+1). Thus we have Mg" 2 pnrr". Now we define a map A": B(G"+1)G€BB(A">A s B((G/A)"+‘)GEBB({A}")A 30 by the formula fi"(f, f’) : (rrnf,/\"‘1f’). Then we have d/Bn(f’ fl) = d(’/Tnf, An_1f’) : (d’fl'nf, _pnflnf _ dAn—lfl) : (fin-de, _Anqnf _ Andfl) = fi"+1(df. -q"f - df') = B"+1d(f, f') and so fl" commutes with the boundary operators. Finally, note that |l5”(f, f')|l(w) = mwfllfinfll, (1+ wl’lll/V'f’lll S mwillflls (1 +W)_1||f'l|} = ||(f,f')l|(W) and so the map 6" has the norm “5"” S 1 for the norm II . |I(w). This shows that ll . “(0) 3 ll - ”C(01). Thus we have n . ”(0) = II - “6(0) on fi*(A s G) for every (0 E [0,00] by passing to the limits. [:1 Notation: We always distinguish a (co)homology class from a (co.)chain by us- ing brackets: for example, [f] stands for a (co)homology class while f stands for a (co)cycle. Theorem 2.14. Let A be an amenable subgroup of G, and let (p: A H G be an inclusion homomorphism. Then, for every n _>_ 2, the induced homomorphism H"(i): H"(A 3) G) —> H"(G) is an isometric isomorphism for the norm II - II, i.e., H "(2) preserves the canonical seminorms. Proof. By Proposition 2.3 and Proposition 2.13, it is enough for us to consider the complexes in the sequences (2.3) and (2.4). We define the complexes M"(A s G) = B((G/A)"+1)G QB({A}A)A EM"(A s G) = B({A}"“)" ®B("“)G EBBMAW. Then the exact sequence 0 s M"(A s G) s EM"(A s G) s B({A}"+1)A s o 31 induces a long exact sequence Since A is amenable, the map H *(i) is an isomorphism. Also it is clear that the map H *(i) has the norm |IH‘(i)|I S 1. We denote by 8,. and a; the boundary operators on B((G/A)*)G and B({A}*)A respectively. Let (f”, f, f’) e B({A}n+1)A a; B((G/A)n+1)" ea B({A}")A be a cocycle. Then, by definition of the boundary operator, we have aiif” = 0: anf : 0) and f” _pnf _ air—1f, = 0 Let n be odd, so that 6;, = id. Then we have f" = 0 and so d(f,f') = ((9nf, -Pnf — 3.2.11“) = 0- It is easy to check that (H"(i))‘1([f”,f,f’]) is represented by a cocycle (f, f’) E M"(A 3+ G) and |I(H"(i))_1HSI. If n is even, then 8,24 = id. So there is an element ff,’ 6 B({A}")A such that as. 5' = f” and urn = llfé’ll- Then we have (f",f,f') — d( 6'00) = (0af,f' —f6') and also d(f, f’ — 6’) = (an,—p"f — 6:,_1f’ + 0;, 6’) = (0,0). Now it is easy to check that (H"(i))‘1([f”, f, f’]) is represented by a cocycle (f, f’ — 6’). Remark that llf' - fé'll = Ilal._1(f' - f3)” = llf" -pnf - 3L4 6'” = llpnfll S llfll- Thus we have ||(H"(i))’1([f",f,f'l)|| S “(f,f' - f6’)|| = max{||f|l, llf' - fé'll} = llfll S max{|lf”|l, llflls Hf'll} = ||(f",f,f')||. This shows that I|(H"(i))‘1II S 1 is also true for every even n. Thus the isomorphism H "(2) is also an isometry. Cl 32 From Theorem 3.8.4 in [8], it is proved that, if A is an amenable normal subgroup of G, then the groups H*(G/A) and H*(G) are isometrically isomorphic. Hence, by Theorem 2.14, the groups H*(A 3) G) and PI*(G/A) are also isometrically isomor- phic. Theorem 2.15. Let A be an amenable subgroup of G and let (p: A ‘—> G be an inclusion homomorphism. Then the norms [I ~ ”((0) on the group H"(A 3) G) are equal for n 2 2 and for every w E [0,00]. Proof. Let b.) > 0. Since it is clear that II - II(w) S II - II = II - “(0) for every (0 E [0,00] from definition, we only show that I] - II(w) _>_ [I - II. By Proposition 2.3 and Proposition 2.13, it is enough for us to consider the com- plex B((G/A)"+‘)G e B({AMA If (f, f’) is a cocycle of the complex B((G/A)"+1)G QB({A}")A, then we have 0 = cm, i) = (as, —p"f — 6:.-.f'). where p": B((G/A)"+‘)G —+ B({A}")A is defined as a restriction map and so it has the norm IIp"II S 1. Also 6 and 8’ are the boundary operators on B ( (G /A)‘)0 and B({A}")A respectively. Recall that, since the group A is amenable, if n - 1 is odd, then 6;“, = id and so —p"f - 8:,_1f’ = —p"f — f’ = 0. Thus f’ = —p"f. This shows that ||(f, f’)|l = max{llf|l, “fl” = mEhflllfll, llp"fll} = Hfll = mA~X{l|fll, (1 +w)_1|lp"f||} = max{||fll, (1 +w)"1||f'||} = ||(f,f')l|(W) and SO lllf, f'lll S “If. f'l|l(W)- On the other hand, if n — 1 is even, then 0;“, = 0. So 8:,_1f' = 0 and so f’ E ker(0;,_1) = Im( :4). Note that 8;,_2 = id. Thus there is an element f” E B(An’1)A such that 8:,_2f" = f’ and IIf”I| : IIf’II. Note that (f, f’) + dn_1(0, f") = (f, f') + (0, -31._2f”) = (f,0)- 33 Thus we have “if. f’lll = ”mom 3 max{||f||,0} = llfll = ||(f.0)ll(w) and so ”If, f'lll S ”If, 0l||(w) = ”If, f'l||(W)- By passing to the limits, we have |I[f, f’]|I S |I[f, f’]II(w) for every (0 E [0,00]. This finishes the proof. El Theorem 2.16. Let (,0: A —> G and (0’: A’ —> G’ be the group homomorphisms respectively. Let a: G —> G’ and 'y: A —-> A’ be the surjective homomorphisms with the amenable kernels respectively and such that a 0 (,0 = (,0’ o 7. Then the groups PI*(A’ 3’) G’) and H*(A 3) G) are isometrically isomorphic for the norm II - II(w) for every (.0 E [0,00]. This isomorphism preserves the canonical seminorms. Proof. Denote ker(a) and ker(7) by K and N respectively. We identify the groups G’ and A’ with G / K and A/ N respectively and denote the homomorphism A /N ——> G / K by p. Then we have a commutative diagram A i A /N A A G —is G/K. Remark that p is defined by the formula p(N a) = K (p(a). It suffices for us to show that H*(A/N A G / K ) and H*(A 3) G) are isometri- cally isomorphic. Since K and N are amenable normal subgroups of G and A respec- tively, the groups H*(G) and H*(A) are isometrically isomorphic with H*(G/ K) and fi*(A/N) respectively. We consider the standard G / K - and A/N-resolutions of the trivial G / K - and A /N -module R. Note that there is a diagram B((G/K)n+l)G __fl_) B(Gn+1)G __7'_"_, B((G/K)"+1)G B((A/N)"+1)A —i"—s B(An+1)A —A"—> B((A/N)"+1)A where each row consists of the maps in Lemma 2.9 such that uni" = id and Anj" = id and also they have the norms IIi”II = 1, ||j"|I = 1, |I7r"|I S 1, and II/\"II S 1. The 34 maps p” and q" are defined by the formulas pnf(a11' ' '1an+1): f(97(a1)1' ' ° 190(an+1)) qnf’(Nal,'°' ,Nan+1) = f,(ch(a1)1"' ,K(p(a,,+1)). We prove that this diagram is commutative. It is easy to check that the first square is commutative. For the second square, we note that qnflnf(Na17' ' '1NaTH-1): an(K(p(0’1)3 ' ' ' 1K90(a"n+1)) = mean off on (K(0(a1),--- ,K(p(a,,+1)) : mean Ofpnf on (Nah... aNan-l-l) = Anpnf(a17 ' ' ' ian-l-l) and so we have anr" = X‘p". From definitions, we have the following complexes M"(A/N 2. 0/10 = B((G/K>"“)G EB B((A/Nr)" EM"(A/N 3> 0/10: B((A/N)"+1)A 69 B((G/K)"+1)G EB B((A/NlnlA M"(A s G) = B(G"+1)G QB(A")A EM”(A s G) = B(AA+1)A Q B(G"+1)G Q B(A")A. We consider the following diagram 0 s M”(A/N s G/K) ——+ EM”(A/N s G/K) ——s B((A/N)")A s 0 e1 , «1 .1 0 s M"(A i G) ——s EM"(A is G) —> B(A”)A s 0 in which each row is exact and each column is defined by the formulas a"(f", f. f') = (J'"f",i"f,j"”lf'), m(f) f') = (i"f,j"_1f')a 7"f” = inf”- It is easy to check that these maps commute with the boundary operators and the diagram is commutative. Also this diagram induces the following commutative dia- 35 gram Err-1mm) —s firm/N s G/K) —> PING/K) —s firm/N) Hn_l(7)I H"(B)I Hulall Hu<7lI Ere-1m) ——> PI"(A s G) ——> fin(G) —> H”(A ). Since H *(a) and H *(7) are isomorphisms, the map H ‘ (fl) induced from ,8“ is also an isomorphism. Also note that we have ”5"“ S 1 for the norm || - ”(10) for every w 2 0. So the induced map H"(B) has the norm IIH"(,B)II S 1 for the norm II — ”(01) for every w E [0, 00]. On the other hand, we define E": M"(A s G) s M"(A/N s G/K) by the formula fi"(C,C’) = (7r"(:,)\"_1C’). Then we have Mme) = d"(r”C, WC) = (d"rr"C. —q"w"< — drums) = (w"+‘d"<. —A"p"c — Mar—‘4’) = B"+‘(d"<, —p"< — d'rlc’) = B"+1d"(c. C’) and so 3” commutes with the boundary operator. It is easy to check that finfi" = id. Since we have IIrr"|I S 1 and [IV—III S 1, it is clear that the map E" has the norm II/T‘II S 1 for the norm || - I|(w) for every (.0 2 0. Hence the induced map H"(B): H"(A 53> G) —> H"(A/N 1) G/K) is the inverse of H"(fi) and also has the norm IIH"(E)II S 1 for the norm I| - “((0) for every w E [0, 00]. Thus the isomorphism H "(5) is also an isometry. Cl Corollary 2.17. Let A be an amenable group, and let (p: A —> G be a group homo- morphism. Then the groups H"(A 3) G) and PI"(G) are isomorphic. Furthermore, the norms II - ”((0) in H"(A 3) G) are equal to the norm II - II in H"(G) for every wE [0,00]. Proof. We note that the image (p(A) is an amenable subgroup of G and also note that ker((,0) is an amenable subgroup of A. 36 We denote by p: (p(A) H G an inclusion map and consider the diagram A —‘P—> 90(4) 1:1 Al G i G. It is clear the diagram is commutative and the horizontal maps are the surjective maps with the amenable kernels respectively. Then, by Theorem 2.16, the groups ff"(A 31) G) and H"((p(A) 3) G) are isometrically isomorphic for the norm II - II(w). Now, by Theorem 2.14, the groups H"((p(A) 3) G) and H"(G) are isometrically isomorphic for the norm II - II. Also, by Theorem 2.15, the norms II - II(w) on the group PI"((p(A) —p—> G) are equal to the norm II - “(0) = II - II. C] In the rest of this chapter, we let A be a subgroup of G and let (,0: A Ls G be an inclusion homomorphism. Then we give another description of relative bounded cohomology of G modulo A. Definition 2.5. Let 6- a a 61 6’ 8’ 0—)R—‘>U0—°>U1—‘>~- and OsR—‘n/o—svl—‘sm be an allowable pair of resolutions for (G,A;R), and let A": U" —) Vn be an A- morphism of resolutions commuting with the contracting homotopies as in Definition 2.1 . If A" induces a surjective map )1": Us; —> K? as the restriction map of A" for every n 2 0, this pair of resolutions is said to be proper. Proposition 2.18. The pair of standard G- and A-resolutions is proper for (G, A; R). Proof. From Proposition 2.1, the standard G- and A-resolutions are an allowable pair for (G,A; R). Note that the map p": B(Gn“) —> B(A"+1) is defined by the formula pn(f)(a1, ' ' ° ran-t1) : f(a’l) ' ' ' ian+1)' It suffices to show that the restriction map p": B(G"+1)G —+ B(A"+1)A is surjec- tive. Note that for every G—invariant element f in B (G"+1) the value of f at every 37 (91,92, ' " agnign-I—l) E G"+1 is independent of gm”. So we can identify a function f E B(G”+1)G with the function f, E B(G") defined by the formula f, (91, ° " 1911) = f (91, ° -- , gm 1). Conversely a function f’ E B(G") is identified with the function f E B(G"+1)G defined by the formula f(g1,--- ,gn,g,,+1) = f’(g1,--- ,gn). Thus the subspace B(G"+1)G of G-invariant elements in B(G"+1) can be identified nat- urally with B(G"), and similarly B(A"+1) with B(A"). Hence the surjectivity of the restriction map, B(G"+1)G —+ B(A"+1)A, follows from the fact that the maps p""1: B(G") —> B(A”) are surjective for all n 2 1. This finishes the proof. [:1 Let 6-1 30 a. 6:1 0;, a; OsR—on—sUl—sm and O—hR———>V0—+V1—>~- be a proper pair of resolutions for (G, A; R). Then there is an exact sequence 0 —> ker(p") —> Us; —> K? —> 0. It is easy to check that the sequence 0 —> ker(p0) —> ker(pl) —> ker(p2) —> (2.5) is a complex. Definition 2.6. The n-th cohomology of the complex in (2.5) is denoted by H"(G, A). As an important example, the standard G- and A-resolutions induces an exact sequence 0 s P"(G', A) s B(G"+1)G ”—"h B(A"+1)A s o, where P"(G, A) = ker(p”). Also this exact sequence induces a long exact sequence s Pin-WA) s ENG, A) s fmG) s IT‘(A) s 38 Theorem 2.19. The groups H"(G, A) and H"(A 5 G) are isomorphic. This isomorphism carries the seminorm on H*(G,A) to a seminorm equivalent to the canonical seminorm on H‘(A 3) G). Proof. Recall that the complexes induced from the standard resolutions M"(A s G) = B(G"+1)G QB(AA)A EMn(A_,G):An+1A@BGn+1G®BAn We consider the following diagram 0 s P"(G,A) —s B(G"+1)G is B(A"+1)A s o fin‘] I an‘I 7nJ’ 0 s M"(A s G) —is EM"(A s G) i B(A"+1)A s 0, where an(f) = (pnf,f,0), 7,,(f”) = f”, and fin(f) = (f,0). It is clear that the diagram is commutative and so there is an induced commutative diagram s Err-1m) ————> f1"(G,A) —s fmG) ———> H"(A) s H"—1(7)I H"(/3)I H"(:)I H"(7)I —> fin‘1(A) —s H"(A 53> G) ——> H"( (G) ———> H"(A) —> Note that the maps H *(oz) and H *(7) are the (isometric) isomorphisms. So the map H*(fi) is an isomorphism. Also, since IIBn(f)II = II(f,0)II = IIfII, the map H*(B) has the norm IIH‘(B)II <1 Let (f, f’) E M"(A —> G) = B(G"+1)G®B(A")A be a cocycle. Then an = 0 and also 8;,_1f’ 2 —p,, f . Since pn_1 is surjective, we can choose an element f1 E B(G")G such that Pn—lfl = f’ and ||f1|| = llf'll- Then pnf : _aii—lf’ : —6h—1pn-lfl : _pnan-lfl so that f + (9,,_1f1 E P"(G, A) and 0,,(f + (9,,_1f1) : 0. Now it is easy to check that (H"(B )) 1([(f, f’ )I) is represented by a cocycle f + (9,,_1f1 E P"(G,A). Then llf + an—lflll S IIfII+IIan—1IIIIflII = ||f||+(n +1)||f'|| $01 + 2) mAXHIfll, llf’H} = (n + 2)|l(f, f')|| 39 This show that, for [f] E H"(G,A), we have 1 n+2 lllflll S IIH"(B)[flll S Will- 40 CHAPTER 3 Relative bounded cohomology of spaces Throughout this chapter, we assume all spaces are connected countable cellular spaces. Recall that the bounded cohomology of a space X, denoted by PI*(X), is defined by the cohomology of the complex 0 —+ B°(X) —+ Bl(X) —+ B2(X) —> --, where B"(X) is a space of real bounded functions on Sn(X) the set of all singular n-simplices. There is a natural norm II - II in B"(X): llfll = sup{|f(0)||0 E Sn(X)} which turns it into a Banach space. Thus in H"(X) there is a seminorm I|[f]|I = inf II f II, where the infimum is taken over all cochains f lying in the cohomology class If] E H‘(X). A continuous map a: U ——> X induces a homomorphism oz“: B*(X) —) B*(U) and the norm of 01* is bounded by one as |Ia*(f)II S “f” for all f E B*(X). Definition 3.1. Let (p: Y —> X be a continuous map of spaces. The mapping cone M "(Y Z) X) and the mapping cylinder EM "(Y 3) X) of cochain complexes induced 41 by (,0 are defined as follows: M"(Y is X) : B"(X)®B"‘1(Y) EM"(Y is X) = B"(V)QB"(X) QBA—1(V). We define the boundary operators on M "(Y 3) X) and EM "(Y 3) X) by the same formulas as in Lemma 2.2, i.e., on EM"(Y 3) X) d(vn,u,,,v,,_1) :2 (dvn, dumvn — /\"u.,, — dvn_1), where A": B"(X) -—> B"(Y) is a cochain map induced by a continuous map (p: Y —> X. Then we have the complexes 0 s MO(V is X) s Ml(Y is X) s MRY is X) s (3.1) 0 s EM°(Y is X) s EMI(V is X) s EM2(Y is X) s . (3.2) Definition 3.2. The n-th cohomology of the complex in (3.1) is called the n-th relative bounded cohomology of X modulo Y and is denoted by H"(Y 3) X). Also the n-th cohomology of the complex in {3.2) is denoted by H"(EM(Y 3) X)). We define the norm II - II on Eh["(Y 3) X) and on M"(Y 3) X) by setting ”(”111 um vn—llll = max{llvnll1llunlla IIvn—III} II(Umvn—illl = max{|lunlls llvn—1||}- Also, for every w 2 0, we define the norm II - ”((0) on M"(Y 3) X) by setting ||(u,,,v.,_1)||= maX{llunll, (1 + wl—lll’Un—illl- Note that there are corresponding seminorms II - II on the groups if *(EM (Y 3) X )) and H*(Y 19+ X) respectively. Also there is corresponding seminorms II - II(w) on H*(Y 3:) X) for every w 2 0. Finally we define these norms II - ”(02) on PI*(Y 53-) X) for all to in the closed interval [0,00] by passing to the limits. 42 Proposition 3.1. Let (p: Y —+ X be a continuous map of spaces. Then the groups H"(EM(Y 3+ X)) and H"(X) are isometrically isomorphic. Proof. Note that there is a map A": B"(X) ——> B"(Y) induced by the map (,0: Y ——> X. Then, as in Proposition 2.3, the natural projection map EM "(Y 3) X) ——> B"(X) induces the isometrically isomorphic groups if "(EM (Y 3) X )) and H"(X). C] Remark that there is an exact sequence 0 s M"(Y is X) s EM"(Y is X) s B"(Y) s 0 and it induces a long exact sequence s PI"(Y is X) s H"(X) s me) s fin+1(Y is X) s . (3.3) Recall that, as shown in Theorem 1.4, the group H*(X) is canonically isomorphic with H‘(7r1X) and this isomorphism carries the seminorm in H*(X) to the canon- ical seminorm in PI“(1r1X). Thus it is natural to consider the relationship between H‘(Y 3) X) and the fundamental groups 11'1X and 1r1Y. Note that from the induced homomorphism (0,: 71'1Y ——> 11'1X , we can define the relative bounded cohomology H*(1r1Y 31;, 11'1X). Remark 3.1. Let 7r: X —> X be a universal covering of X. As shown in Theorem 1.4, the sequence 0 s R s B°(X) s 31(1)) s 82(X) s (3.4) is a strong relatively injective 71'1X -resolution of the trivial 11'1 X -module R. Also the induced map 7r*: B*(X) —> B*(X) establishes an isometric isomorphism B*(X) —> B*(X)"1X , so that the bounded cohomology H "(1r1X ) induced from the resolution in (3.4) coincides with H*(X) as topological vector spaces. 43 Lemma 3.2. Let (,0: Y ——> X be a continuous map of spaces. Let X —-> X and y —> Y be the universal coverings of X and Y respectively. Then the sequences 0—sR—sBO(X) ——> B1(X) ——>B2(X) ——> 0 —> R —s BOO?) —s B102) —> 32(3)) —s (3.2.1) are an allowable pair of resolutions for (11'1X , 1r1Y;R). Furthermore, there is a commutative diagram ( 3. 2. 2) B"(X)“‘X ——-> B((mxsrlw —> B"(X)"‘X *"1 Al ”’1 Bn(y)1r1Y i B((W1Y)n+1)"ly 5 Bn(y)1r1Y, where the maps )1" and p" are induced by a lifting map A: y —> X and (p...: 1r1Y —> 11'1X respectively. Proof. We denote by G and A the fundamental groups 11'1X and 11'1Y respectively. As explained in Remark 3.1, the sequences in (3.2.1) are strong relatively injective G- and A-resolutions respectively. From Theorem 2.4 in [8], the contracting homotopy 0'+—R<— B°(X)(—B1(X) <—~-- is defined by using the cone construction Sn(X) -—> Sn+1(X). By standard calculation, it is easy to check the map A": B"(X) —> B"(y) is an A-morphism and it commutes with the contracting homotopies. Thus the sequences in (3.2.1) are an allowable pair of resolutions for (G, A; R). Now we consider the diagram in (3.2.2) Bn(X)G L", B(Gn+1)G _C_"_, Bn(x)G *1 A1 *"1 Bn(y)A _1’"_, B(An+1)A _1"_, Bn(y)A_ The maps a" and '7“ are defined by the same formulas in Lemma 2.6. Also the maps C" and n" are defined by the same formulas in Theorem 1.4. 44 Since the resolutions in (3.2.1) are an allowable pair for (G, A; R), it follows from Lemma 2.6 that the first square is commutative. On the other hand, let a: An —> )7 be a singular simplex. Also let (an, - -- ,an) E A"+1 be such that o(v,-) E an_,- - - - anyo, where yo is a fundamental set for the action of A on y(see Theorem 4.1 in [8]). Note that, if o(v,) = an_,- - - - any for some y 6 yo, then /\(0(vt)) = Man—s- - ' ° any) = we.-.) - ° ° 90s(an)/\(y)- (3-2-3) Also note that W8 have U"P"f(0) = P"f(aos~- 1an) = f(‘p*(a0)a°°' s90s(an)) and A"C"f(o) = ("f(A o o) = f((,0...(a0), - ~ ,(p..(a,,)), where the second equality follows from the equation (3.2.3). This shows that the second square in (3.2.2) is commuta- tive, so that nnp" = XI". This finishes the proof. 1:] Theorem 3.3. Let (p: Y —-> X be a continuous map of spaces, and let 1,0,: 11'1Y —-> 71'1X be an induced homomorphism. Then the groups PI"(Y 3+ X) and H "(11'1Y 31> 71'1X) are isometrically isomorphic for the norm II - “((0) for every w E [0, 00]. This isomorphism carries the seminorm in PI"(Y 3) X) to the canonical seminorm in H"(7r1Y 3) 1rlX). Proof. Let G and A denote the groups 11'1X and 11'1Y respectively. Recall that the canonical seminorm in H"(A 3) G) is induced by the complex B(Gn+1)G ® B(An)A. Let in: X —) X and n2: 37 —> Y be the universal coverings of X and Y respec- tively. By Remark 3.1, we can identify B"(X) 69 BMW) = B"(X)G GB Brlozr‘. We prove that there are cochain maps <1)": B"(X)G Q BA-1(y)A s B(G"+1)G Q B(A")A and (ya: B(Gn+l)G$B(An)/l ___) Bn(X)G$Bn—l(y)A 45 such that \II"" is chain homotopic to id and they have the norms II"II S 1 and |I‘IJ"II S 1. Recall that there is the commutative diagram in (3.2.2) B"(X)G i"s B(G"+1)G ——‘"—s B"(X)G ”l ("1 ”l Bn(y)A __‘7"_, B(An+1)A i, Bn(y)A so that pna" = 7"/\" and XI" = rynp". Note that, from definitions, the maps ("an and ryny" are chain homotopic to id Esq X)G and idBn(y)A respectively. Also they have the norms”01"” S 1. |IC"H S 1. |I’7"H S 1, and |In"|| S 1. We define CI)" and \II" by the formulas ‘1’"(fsf') = (01"fs7"_1f') and ‘1’"(usU') = (Casi—111'). For simplicity, we denote all boundary operators by the same notation d. Then ‘1’"+1d"(f,f') = ‘1’"“(dfs -/\"f - d"_lf') = (a"+1d”fs -7">s"f - 7"d"‘1f') = (fanfs —P"a"f - 01"“1vn’1f') = d"(01"fs‘r""1f’) = d"<1>"(fs f') and so (1)" commutes with the boundary operators. Also for every w 2 0 ”‘1’"(f, f’)ll(w) = II(a"fs '1"“f')ll(w) = maX{|la"fl|s (1+ w)’1||7""1f'|l} S maX{||f||s (1+w)'1||f’||} = II(fs f’)|l(w)- Thus we have II"II S 1 for the norm II - II(w) for every (0 Z 0. By the same way, we can prove that \I'" commutes with the boundary operators and it has the norm II\I/"|I S 1 for the norm II - “((0) for every (.0 _>_ 0. Finally, from definitions, the map "\II" is chain homotopic to the identity and PI"(Y 3) X) and PI"(11'1Y 3+ 1r1X) are isomorphic. Also, since we have II"II S 1 and I|\II"II S 1 for the norm II - “(1.0) for every (.0 Z 0, these groups are isometric for the norm II - ”(G)) for every (.0 2 0 and so for all (12 E [0,00] by passing to the limits. [:1 46 We recall that the group H*(X) is zero if 11'1X is amenable. Corollary 3.4. Let (p: Y —+ X be a continuous map of spaces such that the fun- damental group rrlY is amenable. Then the groups H"(Y 31) X) and H"(X) are isometrically isomorphic for the norm II - II. Proof. We have the following sequence of isometrically isomorphic groups H"(Y 3+ X) “E H"(1r1Y 3) 11'1X) by Theorem 3.3 ”E H”(1r1X) by Corollary 2.17 E’ PI"(X) by Theorem 1.4. This finishes the proof. Cl Theorem 3.5. Let (,0: Y1 ——> X1 and p: Y2 —+ X2 be the continuous maps of spaces respectively. Let 0:: X1 —> X2 and '7: Y1 ——> Y2 be the continuous maps of spaces such that a o (,0 = p o 7. Let the induced homomorphisms 0...: 1r1X1 -> 11'1X2 and 7...: 1r1Y1 —> 71'1Y2 be the surjective maps with the amenable kernels respectively. Then, for every w E [0,00], the groups H"(Y2 5 X2) and H"(Y1 3) X1) are isometrically isomorphic for the norm II - ”((0). Proof. Let (,0...: 11'1Y1 —> 7r1X1 and p..: 1r1Y2 —> 1r1X2 be the induced homomorphism by (,0 and p respectively. We consider the following diagram 7: 7713/1 ———> 7711/2 (,0.I MI, 11'1X1 _a.__) WIXQ. It is clear that this diagram is commutative and the horizontal maps 7,. and a... are surjective maps with the amenable kernels. Thus we have the following sequences of 47 isometrically isomorphic groups H"(Y2 3) X2) 2 H"(7r1Y2 is 11'1X2) by Theorem 3.3 ’—_‘—’ H"(1r1Y1 fl) 11'1X1) by Theorem 2.16 g H"(Y1 3+ X1) by Theorem 3.3. This finishes the proof. C] In the rest of this chapter, we consider a pair of spaces X and Y C X and denote it by (X, Y). Remark that, as in the ordinary cohomology, there is an exact sequence 0 s ker(p") es B"(X) is B”(Y) s o, (3.5) where p" is defined as the restriction to Sn(Y). We denote ker(p") by P"(X,Y). Then there is an induced sequence 0 s P°(X,Y) s P1(X, Y) s P2(X,Y) s (3.6) which is obviously a complex. Definition 3.3. The n-th cohomology of the complex in (3.6) is denoted by PI"(X, Y). For a pair of spaces (X ,Y), there is a natural norm II - II on P*(X , Y) induced from the norm II - II in B*(X) by the inclusion P*(X,Y) <—> B*(X). Thus there is a natural seminorm II - II in H*(X, Y). Note that the exact sequence in (3.5) induces a long exact sequence s F1"(X, Y) s f1"(X)s firm s FIA+1(X,V) s . (3.7) As we will see in the next theorem, if (p: Y E) X is an inclusion of a subspace Y into X, then the groups H*(Y 2+ X) and PI *(X , Y) are canonically isomorphic. Moreover, as the same theorem shows, the natural seminorms on these groups are 48 equivalent. Nevertheless, there is no reason to expect that these norms are equal. From our point of view, the group H'(Y 2> X) with its seminorm is more natural invariant of a pair of spaces (X, Y) and we consider it to be primary invariant. Theorem 3.6. Let (X,Y) be a pair of spaces and (p: Y —> X be an inclusion map. Then the groups H"(X,Y) and H"(Y 3) X) are isomorphic and the seminorms are equivalent. Proof. From definition, the group H"(Y 3) X) is the cohomology of the complex M"(Y is X) = B"(X) Q B"‘1(Y). Also from Proposition 3.1, we can define the group H"(X) as the cohomology of the complex EM"(Y is X) = B"(Y) QB"(X) QBA-1(V). We consider the following diagram (3.6.1) o s P"(X,Y) —'—s B"(X) ”—"s B"(Y) s o 11 a1 71 0 s M"(Y is X) —’—s EM"(Y is X) —i—s B"(Y) s 0 where a"(f) = (pnf, f, 0) and '7" = id and B"(f) = (f, 0). Then it is clear that this diagram is commutative and it induces the following diagram s fin-1m —> jinx, Y) ——s fin(X) ———> firm s H"“(7)I H"(5)I H"(0)I H"('7)I s Era-W) —s any is X) —s fin(X) ——s firm s. Note that the maps H *(7) and H *(a) are (isometric) isomorphism. So the map H ’(B) is an isomorphism. Since “m(f)” = II(f,0)II = maxlllfllsO} = llflls we have ||fi"|l = 1 and 30 the map H"(fi) has the norm IIH"(fi)II S 1. Let [f, f’] E H"(Y 3) X) and we represent it by a cocycle (f, f’) e M"(Y is X) = B"(X)®B"‘1(Y). 49 For simplicity, we denote every boundary operator by the same notation d. From the definition of the boundary operator, we have 0 = d"(f,f’) = (d"f, —p"f — d"_1f’), so that d"f = 0 and p"f = —d"_1f’. Since the map p"—1 is surjective, we can choose an element f” E B"“1(X) such that P"_1f" = f’ and Ilf'lll = IIf’II- Then p"(f + a""1f")= p"f + p"d"’1f"= p"f + d"‘lp"‘1f” = p"f + a""‘f’ = 0 and so f + dn‘lf” E P"(X,Y). Also d"(f + dn‘lf”) = 0 and so f + d“‘1f” is a relative cocycle. It is easy to check (H "(G))—II f, f’] is represented by this relative cocycle f + dn‘lf”. Remark that llf + d"’1f”l| S llfll + lld"‘1|lllf”ll S llfll + (71 +1)llf'|| S (n + 2) maX{|lf|ls Hf'll} = (n + 2)ll(fs f’)|l and SOII(Hn(fi))—1[f1fllllS(n+2)II[f1f,lII' Thus, for [f] E H"(X,Y), we have 1 — < H" < . ”+2111f111_11 (emu _ 1111111 This finishes the proof. . Cl Corollary 3.7. The groups H*(X, Y) and H*(1r1Y i‘—> 11'1X) are isomorphic and this isomorphism carries the seminorm in H *(X ,Y) to a seminorm, which is equivalent to the canonical seminorm, in H*(1r1Y 31> 71'1X). Proof. From Theorem 3.3, the groups H‘(Y 3) X) and H*(1r1Y 3) 11'1X) are iso- metrically isomorphic. Hence it follows from Theorem 3.6. C] Corollary 3.8. Let (X, Y) be a pair of spaces and let the fundamental group 11'1Y be amenable. Then the groups PI"(X, Y) and H"(X) are isomorphic and the seminorms are equivalent. 50 Proof. Let (0: Y Es X be an inclusion map. From Theorem 3.6, the groups PI"(X, Y) and H"(Y 3+ X) are isomorphic and the seminorms are equivalent. Since 1r1Y is amenable, the groups PI"(Y 31> X) and H"(X) are isometrically isomorphic by Corollary 3.4. Hence the groups H"(X, Y) and H"(X) are isomorphic and the seminorms are equivalent. (:1 51 CHAPTER 4 61 homology of groups We now dualize the notion of relatively injectivity to define 21 homology group of groups. Throughout this chapter, G denotes a discrete group. Recall that R and the B(G") are the important examples of G-modules for the theory of bounded cohomology. Now we introduce another G-module which will be useful for computing the E- homology of groups. Let C,’,(G) be a free R—module generated by the (n + 1)-tuples (g0, - -- ,gn) of elements ofG, with the G-action given by go(go, - -- ,g,,) = (g-go, - -- ,g- gn). We take the (n+1)-tuples whose first element is 1 which represent the G-orbits of (n +1)-tuples. We write such an (n +1)-tuple in the form (1,g1,g1g2,--- ,g1g2---g,,) and introduce the bar notation l91I92I ' ' ' Ignl = (1s91s9192,"' s9192 ' "9a), and define Cn(G) as the free R-module generated by the n—tuples [g1Ig2I - - - I 9"] with the G-action. Since the operation on a basis with an element of g E G yields an element g[g1I - - - I gnI in Gn(G), we may describe Gn(G) as the free R-module generated by all gIgII - - - I gn] so that an element of Gn(G) can be written as a finite sum of the form Zr,g,[g,,| - - . Ig,n] where r,- E R, g, E G. In particular, C0(G) has one generator, denoted by I], so its element is a finite 52 sum of the form Zr,g,[]. We define the 61 norm II - III in Gn(G) by putting ll Zea-[gel -~|91..ll|1 = Z lrsl- Now let Cf,‘ (G) be the norm completion of Gn(G). Thus 0511(0) = {ergs[gill---lg.,,]| Elm] < 00} i=1 is a Banach space with the G-action such that II 9 - cIIl S IIcII1 for every g E G, c E Cf,‘ (G) Hence Gf,‘ (G) is a G—module. Definition 4.1. A surjective G-morphism of G-modules 7r: V -—> W is said to be strongly projective if there exists a bounded linear operator 0: W —> V such that 7r 0 o = id and ”(III S 1. Also a G-module U is said to be relatively projective, iffor any strongly projective G-morphism of G-modules 7r: V —-> W and any G-morphism of G-modules a: U ——> W there exists a G-morphism B: U —> V such that 7r 0 [3 = a and ”,3” S Ho“. The definition is illustrated by the following diagram (4.1.1): =U 1A 1“ V——"——+W Lemma 4.1. The G -modules Cf;1 (G) are relatively projective for all n 2 0 . Proof. Let 7r: V —> W be strongly projective G-morphism of G-modules. We consider the situation pictured in diagram (4.1.1), in which U = Cf,‘(G) and all the rest are given. Let x = 2:17‘191111 E U, where u,- = [g,-,| - - - Igin] and gs]. E G. We define [3 by the formula (3(2 T191111) = 2 7391000111 i=1 i=1 Then rrfi = (1 follows from the following: 00 (X) 776(2 r,g,-u,-) : 7T(Z r,g,-oa(u,-)) = Zr,g,-7r(oa(u,)) i=1 i=1 : £739,001,) = 01(2 73-91-11.) i=1 i=1 53 Also note that for every 9’ E G 00 9’5(: T191111) = g’Zr,g,aa(u,) = 2739,200th i=1 i=1 i=1 00 = B(Z 7‘19’91111) = 18(9' 2: T191111) i=1 i=1 and so B commutes with the action of G. Finally, 00 00 “W: rtgtutllli = II ETiQtUOWillll 00 00 5 Z 1 1 Ilgsnlnennen s E 1 1 Hall, i.e., lll3(:v)||1 S Iltlllllalls for any ft = 22173-9111.- 6 U. Hence [3 is a G-morphism such that it o B = a and IIBII S Hall. El Recall that a G-resolution is said to be strong if it is an exact sequence (as a vector space) of G-modules and G-morphisms which is provided with a contracting homotopy whose norm is less than or equal to 1. Definition 4.2. A strong G-resolution of a G-module V ---—sV2—>V1—sV0—sV—>O is said to be relatively projective if all G-modules Vn are relatively projective. Now we consider the sequence of G-modules and G-morphisms —s C§‘(G) ——> 051(G) —+ Gf‘(G) —> 051(G) —> R —-> O, (4.1) where the boundary operator 8”: Cf,‘ (G) —> C£LI(G) for every n 2 0 is defined by anlgll ' ' ' Ignl = (_1)ngllg2l ' ° ' Ignl 3 —1 + (-1)""[91|-~ Igigi+ll"' lgnl + [film lgn—lls :1 '0. while sII = 1 is a G-morphism e: Cg‘(G) —> R. 54 Also we define s_1: R —> 061(G) and sn: C“(G) —> Cf,;1(G) by the formulas Tl. respectively: s—11=[l and 8n(9[91|-~|gnl) = (-1)"+llgl91|-~lgnl- By the standard calculation and by Lemma 4.1, it is clear that the sequence in (4.1) is a strong relatively projective G—resolution of the trivial G-module R. Definition 4.3. The sequence in (4.1) is called the bar resolution of G. Definition 4.4. For any G-module V the space of co-invariants of V, which is de- noted by Vg, is defined to be the quotient of V by the additive submodule generated by the elements of the form gv — v for all g E G and v E V. For any strong relatively projective G-resolution -~—>V2—>V1—>V0—sR—>O of the trivial G-module R, it is easy to see that the induced sequence —> (V2lG —> (V1)G —* (VolG —> 0 (4-2) is a complex and the homology of this complex depends only on G. Definition 4.5. The n-th homology group of the complex in (4.2) is called the n—th 51 homology group ofG and is denoted by H,‘,‘ (G). Note that the homology of the complex in (4.2) has a natural seminorm which induces a topological vector space structure. Also note that this seminorm depends on the choice of a resolution. Definition 4.6. We define the canonical seminorm in Hf1(G) as the supremum of the seminorms which arise from all strong relatively projective G-resolutions of the trivial G -module R. 55 We shall see the canonical seminorms on Hf‘(G) can be achieved by the bar resolution of G from the following theorem. Theorem 4.2. Let a! a' 8’ I msnsmswsRso (— (— t— <—— 22 ‘1 to {_1 be a strong relatively projective G-resolution of trivial G-module R. There exists a G -morphism of the bar resolution of G to this resolution ——s G§1(G) is Gf‘(G) —""—s 051(G) —E—> R —s o la 13 1A 1A 1A 1 ——sV2——sV1——>Vo—>R—->0 extending idR and such that IIntI S 1 for every n 2 0. Proof. We define fn by the formula fn(g[91| - - ' lgnl) = (-1)"gtn-1(gltn—2(gz - ° - (gushed—1(1)) ' - ' )- It is clear g - fn([g1I - - - IgnI) = fn(g[glI - - - IgnI) so that fn commutes with action of G. Moreover, since |It,.II S 1 and Hg . xII1 S IIxIIl for all g E G,x E K, the map fn is a G-morphism and has the norm |IntI S 1. Note that, for n = —1, we have e’fOI] = 1 2 GI]. It remains for us to verify that fn8n+1 = 8:,+1f,,+1 for every n 2 0. First, note that fn+l (lgll ' ' ' Ign-l-ll) : (_1)n+ltn(gltn—l(g2 ' ' ' t0(gn+1t—1(1)) ' ' ') = (-1)"+1tn(glfn(lg2| - - - l9n+11))- Then we assume fn_1('9,, = (9:,fn and we prove that fn0n+1 = 6:,+1fn+1 by the 56 induction on n. Since 8:,th + tn_18;, = id, we have Bisstfssstflgll Ign-HI): as..(—1)"+1(ts(gsf.(1gsl---19....11» = (-1tn)"+105.+1 (glfn(l92l - - - lgn+1lll = (-1)"“(id - tn_181.)(91fn([92| - - - Ign+1l)) = (-1)"“91fn([92l -lgn+1l) - (-1)"“tn_13l.(91fn(lgzl - - - I9n+11)) = (-1)"“fn(91[92| - - - Ign+1l) + (-1)"tn_1(9131.fn([g2| ' ° - lgn+1l)) = (—1)n+1fn(gl[g2l"'Ign-Hl) + (-1)"tn_1(91fn—13n(l92|~-|9n+1l)) = (-1)"“fn(91[92| - - - Ign+1l) + (-1)"tn—1(glfn—1((-1)"g2[93| - - - last] 71. + Dari-A1921 - - - 19.9...) . - - lash] + 1921 - . - 1g.1)) i=2 = (“1)"+1fn(91l92I ' ' ' lgn+1ll + (—1)"(—1)"t,,_1 (9192fn—1(l93| ' ' ' I9n+1lll + Z(—lln+l_i(—1lntn—191fn—1([92] ' ‘ ‘ I919i+1l ° ° ‘ I9n+1ll + (_1)ntn—lglfn—l(lg2l ' ' ' IgnI) : (_1)n+1fn(gllg2l"'Ign+1l)+(—1)nfn(lglg2lg3l' ' ' Ign+1l) n + Z(-1)"+1“‘fn(l91|92|~- lags-sell --- Ign+1ll + fn([91| - - - lgnl) i=2 : (_1)n+lfn(gll92l ° ' ' Ign+1ll +:Zl(1ln 1fn( [‘91I92I I9191+1I"'Ign+1])+fn([91I"°IgnI) : fn an+1(lgll'” Ign+1l)' Thus we have fn8n+1 = 8:,anH. [:1 Corollary 4.3. In Hf‘ (G) the seminorm induced by the bar resolution of G coincides with the canonical seminorm. Proof. It follows from Theorem 4.2. C] 57 Recall that the ordinary cochain groups of a space are defined as the algebraic dual space of the chain groups. Now we describe bounded cochain groups as the dual space of 81 chain groups. Proposition 4.4. The space B(G"+1)G is the dual space of Cf,1(G)G. Moreover the boundary operator dn_1: B(G")G ——> B(G"+1)G is the adjoint of the operator 0,: C§I(G)G s cfil_,(G)G. Proof. If :1: = 213191,) - - . Ig..,] e C§I(G)G and f e B(G"+1)G, then < £13,f >2 Zrif(gi11”' 1911111) Also for every f E B(G")G < £13,dn_1f >2 Dnzri(dn~lf)(giiv' ' '1gin11) =eri((_ f(giza'” 191.,11) +§—:l(_1 )n jf(gii>”°1gijgij+1a”'agin11))+f(giir'°' agin—ligin '1) j=l p—i "— : Zri((_1)nf(gi21' ' °1gin91)+ (_1)n—jf(gi11' . . ’gijgij-H’. . . ’gin’ 1)) K). II H +f(91'11"'191,_111) =< r1((-1)"9[11| |91..l + Zlghl lg1,g1,.1|~°|g1..l + [91'1I°'°I91'1._1l)1f > =< an,f > . This finishes the proof. Cl Remark 4.1. In [10] Theorem 2. 3 shows that if Im{C ,, +1( )-——> a.“ Cf,‘(G)} is closed in Cf,‘ (G), then H "+1(G) is isomorphic with the dual Banach space of Hf,‘+1(G ) Also it is known that H (1 (G ) = 0. In fact, by using the bar resolution of G, it is easy to see that 81 = 0. Also for any [9] E Cf‘(G)G, it is constructed S([g])=z,:°_2m—1%;T2[g | ngI which is clearly IIS(Ig])II1 = 1 and 823([g]) = [g] showing that Hf‘(G) = 0. 58 Note that H ,9 (G ) is a covariant functor of G. As in the bounded cohomology, this functoriality can also be described in terms of arbitrary resolutions as follows: given a group homomorphism a: G -> H and strong relatively projective resolutions L! and V of the trivial G- and H -modu1e R, respectively, we can regard V as a strong G- resolution via 0. So we have an augmentation preserving G-morphism of resolutions 7': u ——> V, well-defined up to homotopy. The condition that r be a G—morphism is expressed by the formula r(gx) = a(g)r(x) for g E G and x, E Ll. Clearly r induces a map UG —+ VH, well-defined up to homotopy, hence we obtain a well-defined map 0...: Hf‘ (G) —> Hf1(H) which depend only on 01. Note that I|oz,.I| S 1. Now we shall see the relationship between amenable groups and 81 homology. Let A be an amenable subgroup (of G. We consider G /A, the set of (right) cosets Ag of A in G. Since the set of cosets Ag has the G—action by right translation, we can define Cf‘ (G /A) as the same manner with Gf1(G). Namely, we can take Gf,‘ (G /A) as the free R-module generated by the n-tuples of the form [Agll - - - IAgn]. The action of a G-module is given by the formula 9’ [AgII - - - IAgn] 2 Ag’ [Agll - ~ - IAgn]. The canonical map pn: Cf,‘(G) -—> Cf,‘ (G/A) is a G-morphism and has the norm IIpnII S 1. Lemma 4.5. Let A be an amenable subgroup of G. Then there exists a G-morphism q1: Gf1(G/A) —> Gf‘(G) such that pl 0 £11 2 id and IIqIII = 1. Proof. Recall that there is a right invariant mean on B (A), i.e., the linear functional m: B(A) ——> R so that m(a-f) = m(f) where a-f(a’) = f(a’a) for a,a’ E A and f E B (A) Also recall that, on any coset Ag, the map m defines a mean mg: B(Ag) —> R by met/2) = m(f), where f(a) = 10mg). For each x E G, consider the characteristic function 6x: G —) R, i.e., 1 if y = x, 622(31) : 0 otherwise. 59 For every Ag’ [Ag] E Cf‘ (G /A), we define ql by the formula (1(1A9 [A9]): 27719051: IAg) glxl' xEG Since 0 S (LI/,9 S 1, we have 0 S mg(6$IAg) S 1 and also Z mg(6,,IAg)I = ngwxlAg) 2mg (:6 IAg)= mg( A9) = 1, (4.5.1) xEG xEG 360 where 149 is a constant function on Ag with value 1. Thus we have IIql II = 1. Note that p10q1(][Ag)=p1((ng(1-5IAg))[$I xEG = ng(61|A,)p1([r1) = Z mg(6.|,,g)p1([r)) xEG xEAg = Z "1164.101149] = (Z mg(51|Ag))[Ag] $6149 xEG = [A9], where the last equality follows from (4.5.1). This shows pl 0 (11 = id. Finally note that g’qr([Agl) = g' Zm1(6.|,,,)1e1 = Zm1161|,,)g'1e1 ' 260 36G = q(A9'[Agl) = q(9’[Agl) and so ql commutes with the action of G. [:1 Corollary 4.6. Let A be an amenable subgroup of G. Then for every n 2 0 there exists a G-morphism qn: Cf,‘(G/A) —> Cf,‘(G) such that pnqn = 1 and IanII 2 1, where pn: Cf,‘(G) —> Cf,‘(G/A) is the canonical map. Proof. Since the spaces G61 (G) and G61 (G /A) have only one basis element denoted by H, we define qo by the formula q0([]) = I]. 60 Note that (G /A)" = Gn/A" and A” is an amenable normal subgroup of G". We may consider Cf,‘(G/A) as Cf‘(G"/A") by setting up each basis [AgIIn-IAgn] of C§‘(G/A) by [A191] ' ' ' lAgnl = A"l(911“-1911)l- Then Lemma 4.5 provides a Gn-morphism q": Cf,‘ (G /A) -+ Cf;l (G) such that pnqn = 1 and IanII = 1. Especially, since the G-module structure on Cf,‘ (G) and on Cf,‘(G/A) are the restriction of the canonical Gn-module structure, pn is also a G-morphism. Cl Lemma 4.7. Let A be an amenable subgroup of G. Then every G-module Cf,‘(G/A) is relatively projective for every n 2 0. Proof. Consider the diagram 03(0) -—p—> C§‘(G/A) Al 1: V —"—s W where a G-morphism a and a strongly projective G-morphism it are given. We need to construct a G-morphism B: Cf,1 (G /A) —> V such that n5 = a and II G“ S IIaII. Since C,’,1(G) is a relatively projective G-module, there exists a G—morphism 3’: Cf,‘(G) —s V such that ap = rrfi’ and “5’“ S IIapII S IIaII. Moreover, there is a G—morphism q: Cf,‘ (G /A) —> Cf,l (G) constructed in Corollary 4.6. We define B = fi’q. Then M3 = afi’q = apq = a and also ”fill = llfi’qll S llfl’lllIQIl S HOIIIIQII S llall- 13 We introduce another useful strong relatively projective G—resolution. Let A be an amenable subgroup of G. From Lemma 4.7, the G—module Cfl (G /A) is relatively projective. Moreover the sequence s 051 (G/A) is 051 (G/A) is Gf‘(G/A) is 051(G/A) s R s o (4.3) is a strong relatively projective G-resolution of the trivial G-module R, where the boundary and contracting operators are defined by the same formulas in the sequence 61 (4.1). Also it is easy to see that the induced sequence s (ism/As; is (rm/Ase is chm/14).; is Calm/Aw is o (4.4) is a complex and the homology of this complex is Hf‘ (G). Proposition 4.8. Let A be an amenable subgroup of G. Then the seminorm in Hf‘ (G) induced by the resolution in (4.3) coincides with the canonical seminorm. Proof. Let II ° “1 denote the canonical seminorm in Hf1(G) and II - II; the seminorm in H51 (G) induced by the resolution (4.3). By definition of the canonical seminorm in 1151(0), we have 11-111 5 11-11.. Note that, as we proved in Corollary 4.6, for every n 2 0 there exist a G-morphism qn: G,f,I(G/A) —> G,’,‘(G) such that pnqn = id and IanII 2 1, where pn: Cf,‘ (G) -> Cf,‘ (G /A) is the canonical map. Thus the seminorm achieved by the resolution (4.3) is not less than the canonical seminorm, i.e., II - III S II - Ili- Hence II - HI = II - II‘]. El Theorem 4.9. Let A be an amenable normal subgroup of G. Then the map (p..: Hf‘ (G) —> Hf‘(G/A), which is induced by the canonical map (p: G —-> G/A, is an isometric isomorphism, i.e., the isomorphism preserves the canonical seminorm. Proof. Note that the sequence (4.3) is the bar resolution of G /A. So the homology of the complex in (4.4) is Hf‘ (G /A) and the induced seminorm in which is the canonical one. Hence it follows from Proposition 4.8. C] As before, we denote the coset A in A by {A}. If A is an amenable group, we have a complex (1 03 r, 35 11 51 e1 36 -+ C3 ({A})A —> Ce ({A})A —> 01 ({A})A —> Co ({A})A —> 0 (4-5) induced from the strong relatively projective A-resolution of the trivial A-module R —> 0.511111) —s 0511114)) is (rm/1}) —'> 0511141) s R —s o. 62 It is clear that the homology of the complex in (4.5) is H f1 (A) and the norm induced by this complex coincides with the canonical seminorm. Remark that the boundary operators in the complex (4.5) are in fact id if n is even, 8:, = 0 if n is odd. Theorem 4.10. If the group G is amenable, then the group Hf1(G) is zero. Proof. It follows from the complex in (4.5) by setting up G = A. [3 63 CHAPTER 5 Relative 61 homology of groups Let go: A ——> G be a group homomorphism. As in the relative bounded cohomology, we define the relative 61 homology of G modulo A. Remark that there is an induced homomorphism 99.: H f1 (A) —> Hf‘(G) which depends only on (,0. Also remark that llsosll S 1- Definition 5.1. Let go: A —-> G be a group homomorphism. A strong relatively pro- jective G-resolution of a G-module U 6 6 6 8 {—- <—- {—- (— k2 *1 k0 k—l and a strong relatively projective A-resolution of an A-module U o' o' o' 6’ m—3+V2—3>V1—1>V0-3>U—>0 (— <— <— <— t2 £1 to t_1 are called a co-allowable pair of resolutions for (G, A; U) if idU can be emtended to an A-morphism of resolutions An: Vn —> Un such that An commutes with the contracting homotopies kn and tn for all n 2 0. Proposition 5.1. Let go: A —> G be a group homomorphism. The bar resolutions of G and A are a co-allowable pair of resolutions for (G, A; R). 64 Proof. Since the proof is very similar to an allowable pair of resolutions in Proposition 2.1, we refer to it in detail. Recall that the bar resolutions of G and A . is 051(0) is 09(0) is 051(0) is R—s o ‘E i: ‘1..— T: a 1 a; , a; 1 as -:C§(A):Cf(A):CS(A)(:R—>0 t2 £1 to t_1 are strong relatively projective. We define a map An: Cf,‘ (A) —> 051 (G) by the formula An([a1|-~Ian])=[so(a1)|---lso(an)]- Then, by the standard calculation, it is easy to check that An is an A-morphism extending idR and such that it commutes with the contracting homotopies. El Definition 5.2. Let (0: A —+ G be a group homomorphism. Let -«-—>U1—+UO—>R—s0 and --s—>V1—>Vo—>R——>O be the G- and A-resolutions respectively such that they are a co-allowable pair for (G, A; R). The mapping cone and mapping cylinder of chain complexes induced by (,0, respectively, are defined as follows: Cn(A If} G) : (Un)G $(Vn—1)A mm is G) = (m 69mg Elam—as Lemma 5.2. Let ---—sU1—>UO—sR—>0 and ---——>V1——>V0—sR—>O be a co-allowable pair for (G,A;R) as in Definition 5.2. Also let An: V" —> Un be an A-morphism of resolutions commuting with the contracting homotopies. Then the sequences 65 .. is02(Ais0) is01(Ais0) is00(Ais0) —so (5.2.1) .. is E02(A is 0) is 1301(A is 0) is ECO(A is 0) —s 0 (5.2.2) are complexes, where the boundary operators are defined by the formulas dn($n7 arr-1) 2 (811331; + An—lan—la —8:;_1an—1) I I dn(ana III”, an—l) : (anan — an—l) anxn + An-lan—la —an_1an—1)- Proof. We check den = 0 for 0,,(A is 0). dn—ldn($na an—l) :: dn—1(anxn + An—lan—ls —a:;_1an—1) : (an—1871:1271 + an-lAn—lan—l _ ATE—18:1—1an—1) a;_2a:1_lan—1) = (0,0). By the same way, we can check dn_1d,, = 0 for ECn(A £> G). C] Definition 5.3. The n-th homology of the complex in (5.2.1) is called the n-th relative £1 homology of G modulo A and is denoted by Hf,l (A 3+ G). The n-th homology of the complex in (5.2.2) is denoted by H,‘,I(EG(A 3) G)). We define the norm II - II] on Cn(A 3) G) by setting II(xnsan—1)II1= II$n||1+Ilan-1lll and similarly on ECn(A 3) G). Remark that these norms define the seminorms || - II1 in Hf‘(EG(A 56-) G)) and in Hf,‘(A 3) G) respectively. Furthermore, for every w 2 O, we define a norm |I - II1(w) on Cn(A 3 G) by putting “(13m an-1)||1(W) = ||$n||1+(1+W)||an—1||1- 66 Then we have the corresponding seminorm II - II1(w) in H 5‘ (A i) G). Finally we define these norms in Hf‘(A 3+ G) for all to in the closed interval [0,00] by passing to the limits. Remark that, for 0 3 col 3 tag, we have II ' ”I = II ' ”1(0) S II ' “10411) S H ' “10412)- Theorem 5.3. Let (p: A ——> G be a group homomorphism. Then the natural inclusion map pn: C£1(G)G ——> ECn(A is G) = G,‘,‘(A)A$Gf,‘ (G)G®Cf,‘_1(A)A induces an isometric isomorphism Hn(p): Hf,‘ (G) —) Hf,‘(EC(A 3) G)) for the norm |I - III. Proof. We consider the exact sequence 0 —* climb is E0..(A is 0)=0,‘,1(A)@05,1(0)G@0,{1_1(A _>Ce1(A)A®C,€I_(A)A—>O. It is easy to check that C£I(A)A$C,€‘_1(A)A is a complex. If (a, b) is a cycle of the complex Gf,‘(A )A @CfiL (A )A, then we have 0 = d(a, b) = (da — b, —db) and so b 2 da. Thus (a, b) = d(O, —a) is a boundary. This shows that the groups Hf‘(G) and Hf1(EC(A 3+ G)) are isomorphic. We denote every boundary operator by the same notation d. We consider the diagram G5;1(G)a —s G£1(A )$G£1( G) esciu —i'—s G£1(G)a dl dl dl Gill—MGM 31* Cilil—I(A)A®CvliI—I(G)G®Cfil—2(A)A 1111+ Off—MGM where fin(a, x, b) = x+Ana and An: Cf,‘ (A) A —> Gf,‘ (G )G is an induced homomorphism from (p. It is clear that finpn = id and that the first square is commutative. Since we have IIpn(x)I|1 = II(0,x,O)II1 = IIxIIl, the map Hn(p) has the norm IIHn(p)II S 1. 67 Note that p},d(a, x, b) = 5,,(da — b, dx + /\b, —db) = dx + /\b + A(da — b) = dx + Add 2 d(x + Au) 2 d§n(a,x, b), so that 5,, commutes with the boundary operators. Also note that |l5n(as$sbllll = Hi + Analll S l|$||1+|la||1 S ||a||1+ ||$||1+||b||1 =|I(aswsb)||1 and so the map [7,, has the norm II 5,,“ S 1. This shows that the inverse map (H,,(p))_l has the norm II(H,,(p))‘1 II S 1. Hence the isomorphism Hn(p) is also an isometry. El Let (p: A —> G be a group homomorphism. Let the sequences ---——sU1—on—sR—>O and s-~—>V1——sV0—>R—>0 be a co-allowable pair of resolutions for (G, A; R). Then there is an exact sequence 0 —s ((4,). is E0,,(A is 0) is 0,,(A is 0) —s 0, where in and pn are natural inclusion and projection maps respectively. Also this sequence induces a long exact sequence —s Hf,‘(A) —s Hf,‘(G) —s Hfi1(A is 0) —s Hf,‘_,(A) —s Remark that the seminorm in Hfl(EC(A 3) G)), which is induced by the bar resolutions, coincides with the canonical seminorm. Also remark that a seminorm in H f1 (A 31> G) depends on the choice of a co—allowable pair of resolutions for (G, A; R). We define the canonical seminorm in H f1 (A 3) G) by the supremum of the seminorms which arise from every co—allowable pair of resolutions for (G, A; R). Theorem 5.4. The seminorm II - ||1(w) in Hf‘(A 31) G), which is induced by the bar resolutions of G and A, coincides with the canonical seminorm for every w E [0, oo]. 68 Proof. Let --s——>U1—>U0—sR—>0 and ---—>V1——>V0—sR—>O be a co-allowable pair for (G, A;R) as in Definition 5.2. Also let An: Vn —) Un be an A-morphism of resolutions commuting with the contracting homotopies. Let a,,: G£I(G)G —> (Un)G and 7,,_1: Cfi‘__1(A)A —> (Vn_1)A be the maps defined in Theo- rem 4.2. We define a map in: C£’(G)G$C£11(A)A —+ (Un)G€B(Vn-1)A by the formula 6,,(33, a) = (aux, e/n_1a). Since we can prove that the map 6,, is a chain map as the same way in Theorem 2.7, we refer this proof to it. Now note that, for every w 2 0, ||5n(fvs a)||1(w) = II(anxs 7n—10)II1(W) = llan$||1+(1+ w)|l7n-1a||1 S ||$||1 + (1 +W)||a||1 = ”(is a)||1(w) and so ”6,,” g 1 for the norm II - II1(w). 1:] Lemma 5.5. Let A be an amenable subgroup of G. Then the sequences —s C§1(G/A) is 051(0/A) is 0f1(0/A) is 051(0/A) is R —s 0 a; 1 a; 1 a; 1 e —> C§‘({A}) —> 05 ({A}) —> Cf ({A}) —> 5({A}) —> R —> 0 are a co-allowable pair of resolutions for (G, A; R). Proof. We define a map An: Cf,‘ ({A}) —> Cf,‘ (G /A) by the formula An([{A}|-~|{A}l) = [At-1M]- The rest of the proof is exactly same as the proof of Proposition 5.1. D 69 Theorem 5.6. Let A be an amenable subgroup of G, and let (p: A <—> G be an inclusion map. Then the seminorm II - II1(w) in Hf‘ (A 3+ G), induced by the complex 0,,(Ais —s0).—_ 0f,1( ()0/A )G@0;1,( {A}) coincides with the canonical seminorm for every 0.) 6 [0, 00]. Proof. Let II - II1(w) denote the canonical seminorm on Hf‘ (A 3+ G) and let II - II‘i(w) the seminorm on Hf‘ (A f) G) induced by the complex Gn(A 3) G). By definition of the canonical seminorm, we have II 1 IIi‘(w) S II - II1(w). So it is enough for us to show that II - |I1(w) S II - IIi’(w). From Theorem 5.4, the canonical seminorm on H f1 (A 3+ G) is induced by the bar resolutions of G and A, i.e., by the complex Cf,‘ (G)G ® CfiLl(A)A. Recall that, from Corollary 4.6, there are maps (In: C“Ci/Ala —> 051(Glo and (1:11 021({AllA -+ Off—MAM- We define the map s,:0§1(0)/A G@C,’,1_({A})A —s 051063)0f,1_,(A by the formula 6,,(x,a) = (qnx, qI,_1a). By the standard calculation, it is easy to check the map 6,, commutes with the boundary operators. Also, since we have IIq,,II S 1 and IIqI,_1II S 1, it is easy to check that 6,, has the norm “6,,“ S 1 for the norm II - II1(w). This shows that H - l|1(w) S H ' lli(w)- Cl Theorem 5.7. Let A be an amenable subgroup of G and let (p: A ¢—> G be an inclusion homomorphism. Then, for every n 2 2, the groups Hf,‘ (G) and H£1(A 3) G) are isometrically isomorphic for the norm II - III. 70 Proof. It is enough for us to consider the following co—allowable pair of resolutions for (G, A; R): —s 0§1(0/A) is 051(0/A) is 0f1(0/A) is 051(0/A) is R —s o —s 0§1({A}) is0§1({A}) is 0f1({A}) 1 —ng1({A}) is R —s 0. Let 0 (Ais —0‘1(0)/A @051, ({A)} EG..(A —s G) = Gf.1({A})A 61905.1 (G/A) (3690:: {(A)} Then there is an exact sequence 0 —s 051({A})A i‘s E0,,(A is 0) is 0,,(A is 0) —s 0 and it induces the following exact sequence .- —s Hf,‘(A) —s 1151(0) iils Hf,‘(A is 0) —s Hf,1_,(A) —s Since the group Hfl(A) = 0, the groups Hf,1(A £> G) and H£1(G) are isomorphic. We denote by 0., and 8f, the boundary operators on C,‘,1(G /A)G and C,‘,1({A})A re- spectively. Note that the induced map An: Cf,‘({A})A —> Cf,‘ (G /A)G is defined as an inclusion map. Let (b, x, a) E C,‘,1({A})A®C,‘,‘(G/A)G @CfiL 1({A})A be a cycle. Then, by definition of boundary operator, we have (93b — a = 0, 8,,2: + a = O, and 8:,_1a 2 Let n be odd. Then (9;, = id and so a = 0. Also since 8;, = O and BLb—a = 6gb = 0, there is an element 0 E Cf,‘({A})A such that 81,0 = b. So we have (b, x, a) = (b, x, O) = (O, x, O) + d(c, 0, 0) and d(O, x, 0) = 0. Thus (Hn(p))'1Hn(p)(lbs$sal) = (Hn(P))_1([$sal) =(Hn(p))‘1([fvs0l)= [0,:c,0] =[bs 1311]- 71 Then II(H.(p))-1(x.am1 = (no, 2:, Olll 5 His, allll and so I)(H..(p))-1n s 1 for every odd n. Let n be even. Then 8;, = id and (9L, 2 0. So a E ker(8,’,_1) = Im(8fi,) and so there is an element 0 E C£‘({A})A such that 8:,0 = a and Hell] 2 IIaIIl. Note that d(O, x + c, 0) = (0, 8,,x + 8,20, 0) = (0, 0, 0) and also (0, x + c, O) + d(O, 0, —c) = (c, x, 61,0) = (b, x, a). Thus (H,,(p))‘1([x, a]) is represented by a cycle (0, x+c, O) E EC,,(A 3) G) and also II(Hn(p))"1(lxsal)|l1Sll(0s=v + c, 0)|l1 = Ila? + CH1 S |I$III+IICH1= II$II1+II01I|1 =II(1‘sa)|I1- This shows that II(H,,(p))‘1II S 1 for every even n. D Theorem 5.8. Let A be an amenable subgroup of G and let (,0: A <—> G be an inclusion homomorphism. Then the norms II 1 II1(w) in Hf,‘(A 3) G) are equal for all n 2 2 and for all to E [0, 00]. Proof. Let w 2 0. Since we have the inequality II - “1 = II - II1(0) S II - II1(w), it is enough for us to show that II - II1(w) S II - II1. Let (x,a) E Cfi1(G/A)G®C,€‘_1({A})A be a cycle. Then we have 0 = d(x,a) = (8,,x + An_1a, —8’_1a) and so 8;,_1a = 0, where A": Cf,‘({A})A —> Cf,‘(G/A)G is an induced map from (p and 6,, and 0;, are the boundary operators on Cf,‘(G/A)G and Cf,‘ ({A}) A respectively. Recall that, if n — 1 is even, then BL, = id and so 6;,_1a = 0 gives a = 0. Thus |I($s0)||1(w)= |I~T||1+(1+W)||a||1=||$||1=||€C||1+||a||1=||($sa)||1- If n — 1 is odd, then 8;, = id and 8;,_1a = 0 gives a E ker(8:,_1) = Im(élj,). So there is an element an E Cf,1({A})A such that Baa“ = a and IIa,,II1 = II8;,a,,II1 = IIaIIl. 72 Then we have (x, a) + d(O,a,,) = (x, a) + (Anon, —8;,a,,) : (:1: + Ana,“ (1 — (9,261"): (:13 "l" A11017130) So we have IllisalII1(W)SII(1/‘ + )‘nam 0)II1(W) = “17 + AnanIIl S II$II1+IIAnanII1 S IIICIII + Ilanlll =||$||1+l|a||1= II(rcsG)I|1- In the both cases we have IIIx, aIII1(w) S IIIx, aIIII. Then, by passing to the limits, we have IIIx, aIII1(w) S IIIx, aIII1 for all to E IO, 00]. [:1 Theorem 5.9. Let go: A —> G and (0’: A’ —+ G’ be the group homomorphisms respectively. Let a: G —> G’ and 7: A —> A’ be the surjective homomorphisms with the amenable kernels respectively and such that a o (p = (p’ o 7. Then the groups Hf1(A’ is G’) and Hf‘(A i”—) G) are isometrically isomorphic for the norm II - II1(w) for every w 6 I0, 00]. This isomorphism preserves the canonical seminorms. Proof. Denote ker(a) and ker(’y) by K and N respectively. We identify the groups G’ and A’ with G / K and A / N respectively and denote the homomorphism A /N —> G / K by p. By considering the following complexes 0,,(A/N is G/K)= 0f,1( (G)/K 0690‘: , (A/N) EC,,(A/N is G/K): 0f,1( (A/N) €919? (G/K) 6305,1_( , (A/N) 0,,(A is 0) =05,1(0 )GEBC,‘,1_(A EG.(A is G) = G5.1(A)A $G:1(G)GQBG.€L.(A)A. since the proof is very similar to the one of Theorem 2.16, we leave it to the reader. Cl 73 Corollary 5.10. Let A be an amenable group, and let (p: A —> G be a group homo- morphism. Then the groups Hf‘(A 3) G) and Hf‘(G) are isomorphic. Furthermore, the norms II - II1(w) in Hf1(A f) G) are equal to the norm II - ”1 in H"(G) for every we [0,00]. Proof. As in Corollary 2.17, the image (p(A) and ker((p) are the amenable subgroups of G and A respectively. We denote by p: (p(A) ‘-—> G an inclusion map and consider the diagram A —i-> (0(14) rl Pl 0 is 0. Then it follows from Theorem 5.9, Theorem 5.7, and Theorem 5.8. Cl Now, for a subgroup A of G, we give another description of relative 81 homology of G mod A. Definition 5.4. Let the following G - and A-resolutions respectively ---—>U1—>U0—sR—+O and ---—>V1—>Vo—>R—>O be a co-allowable pair for (G, A; R). This pair of resolutions is said to be co-proper if an A-morphism An: V,, —> U,, induces an injective map /\,,: (V,,)A —> (U,,)G. Proposition 5.11. The bar resolutions of G and A are co-proper. Proof. Note that the injective homomorphism A <—> G induces the A-morphism Cf1(A) —> Gf‘(G) which is clearly injective. Also, it is easy to check that the in- duced map Cf1(A)A —> Cf‘(G)G is injective. [I] Let ---—sU1—on—>R——)O and --s—>V1—>V0—sR—>O be the sequences in Definition 5.4. Note that there is an exact sequence 0 —) (V,,)A ‘—> (U,,)G —) (Uan/(Vn)A —> 0. (5.1) 74 It is easy to check that the induced sequence —* (U2la/(V2lA _)(U1)G/(V1)A “i (Uolc/(VolA —> 0 (5-2) is a complex. Definition 5.5. We denote by H,‘,‘(G, A) the n-th homology of the complex {52) Note that the sequence (5.1) induces an exact sequence —s 11:1,,(0, A) —s H§1(A) —s 1151(0) —s Hf,‘(G,A) —s H,‘,1_,(A) —s . (5.3) As an important example, the bar resolutions of G and A induces an exact se- quence o —s 0f1(A),, 1'—'s 051 (0)0 i:s 051(0)G/051 (A), —s o. (5.4) We denote the quotient space Cf‘(G)G/Gf‘(A)A by Cfl(G,A). Remark that there is a complex ... _, 051(0, A) —> 0f1(0, A) —s 051(0, A) —s o. (5.5) In [7], for every 0 2 0, Gromov defines a norm II - “1(6) on Hf‘ (G, A) as follows: first we define a norm II - ”1(6) on Cf‘(G)G by putting llill1(9) = ll$l|1+ 9ll€9$ll1- Then, using the quotient homomorphism p...: Cf1(G)G —> Cf1(G,A), we define the norm IIEII1(6) of E E Cf‘(G, A) by taking all 0 E p:1(5) C Cf‘ (G) and setting ll5ll1(9) = inf{ llCll1(9) lPs(C) = 5}- Then there is a corresponding seminorm II - “1(6) on Hf‘ (G, A). Finally we define this norm on Hf‘ (G, A) for all 0 in the closed interval [0,00] by passing to the limits. Theorem 5.12. Let (p: A s—s G be an inclusion homomorphism. Then the groups Hf,1(A 3) G) and Hf,‘(G,A) are isomorphic and the norm II - II1(w) in Hf,‘ (A 3+ G) and the norm II - “1(6) in Hf,‘ (G, A) are equivalent for w = (9 6 [0,00]. 75 Proof. Recall that G..(A is G) = 0:1(G)a ®G:L.(A)A E0,(A is 0)—_——051(A)€B051(0)G$051_,m We consider the following diagram 0 —s Cf,‘(A)A —'2——s EC,,(A is 0) is 0,,(A is 0) —s 0 1~l A A 0 —> Cf,‘(A)A Ls 051(G)G is Cf,‘ (G, A) —s 0 where 7,,(a’) = a’, 0,,(a’,x,a) = x, and 6,,(x,a) = x + Cf,‘(A)A. It is clear that the diagram is commutative and so there is an induced commutative diagram —s H51(A) —s 1151(0) —s H51(A is 0) ———s H5: 1(A) —s Hn(7)l [inhaljv Hn(5)l Hn—1('Y)‘I( —s H51(A) ——s H51(0) ——s H51(0, A) ————s H5: 1(A) —s Note that the maps H...(*y) and H,(a) are (isometric) isomorphisms. So the map H,,(6) is an isomorphism. Let U.) = 0 2 0. Let (x,a) E C£’(G)G®C£L1(A)A be a cycle. Then d(x, a) :- (6x + a, —6’a) = 0 and so 6x = —a. Then we have ll5n($sa)ll1(9) = lli + Csi‘(A)All1(9l S ll$|l1(9) = llill1+ 9ll3$ll1 S llilll + (1 +11))llall1 =ll(1‘sa)ll1(W)- On the other hand, let x E Cf,‘(G)G be a relative cycle so that 6x 6 C,(,‘_1(A)A. Then we have (x, ~6x) E C£1(G)G®C,€L1(A)A and also we have d(x, -—6x) = (6x — 6x, 6’6x) = (0, 0). It is easy to check that (Hf,1 (6))‘1[x] is represented by a cycle (x, —6x). Also note that ll($s-5n$)ll1(W)=llill1+(1+w)ll3n$ll1=llill1+llanwll1+w|l3n$ll1 S (n + 2)llflill1+ cullancvlll S (n + 2)(llill1+ 9ll<9n$llll = (n+2lllill1(9)- 76 Hence, for [:c,a] E Hf,‘(A is G), we have 1 n+2 lllxsal||1(w)S llHn(fi)([-’vsal)||1(9) S llll‘sallldwl 77 CHAPTER 6 Relative 61 homology of spaces Throughout this chapter we assume every space is a connected countable cellular space. For a space X, we denote by C...(X ) the real chain complex of X: a chain c E C. (X) is a finite combination 21.7310,- of singular simplices o,- in X with real coefficients 73-. We define the fll-norm, denoted by H - H1, in C...(X) by setting ||c||1 = Z, |r,-|. Let 05‘ (X) be the completion of C..(X) with respect to this norm, i.e., oo oo Cf‘ (X) = {Zn-oil Iril < 00}. 1'21 i=1 it is easy to check the sequence —s C§‘(X) 513) C§‘(X) iis 0f1(X) i‘s 031(X) —s o (6.1) is a complex of Banach spaces and bounded operators, where the boundary operator 0,, is defined by extending linearly the boundary operator on the ordinary chain complex C..(X) and it has the norm ”a,” S n + 1. Definition 6.1. The n-th homology of the complex in (6.1) is called the n-th 31 homology ofX and is denoted by H£1(X). We define a seminorm ||[x]||1 of [x] 6 H900 by taking all cycles c E Cf1(X) lying 78 in the homology class corresponding to [x] and setting ||[$l|l1 = inflllClll I [11“] = {Cl}. In [10], it is shown that the bounded cohomology f1 *(X ) is zero if and only if the 61 homology Hfl (X) is also zero. Thus, if a space X is simply connected or its fundamental group 1r1X is amenable, then H51 (X) is zero. In the next theorem, we prove that the Hf‘ (X) also depends only on its funda- mental group 71'1X as the group FI‘(X) does. Theorem 6.1. The group Hf‘ (71’1X ) is canonically isomorphic with H51 (X) This isomorphism carries the canonical seminorm in Hf‘ (1r1X) to the seminorm in H51 (X). Proof. Let 7r: X —+ X be the universal covering of X so that 1r1X acts freely on X and X /1r1X = X. The action of «IX on X induces the action on the chain groups Cf‘ (X) and thus turns them into bounded 1r1X -modules. We show that these 1r1X -modules are relatively projective. Let X 0 C X consist one element from each 1r1X-orbit. The complex C...(X) is free on all simplexes the first vertices of which are in X 0 (see Theorem 10.20 in [12]). Then by a standard calculation as in Lemma 4.1, it is easy to see that these 1r1X-modules Cf‘ (X) are relatively projective. Now we consider the sequence —s 05(1)) —s Cf‘(X) —s 051(X) —s R —s 0. (6.1.1) Since X is simply connected, Hf‘ (X) = O and hence the sequence in (6.1.1) is exact. Thus the sequence (6.1.1) is a strong relatively projective 11'1X -resolution of the trivial 1r1X -module R, where the fact that this resolution is strong is shown in Theorem 2.4 from [8]. Remark that the map 7a.: Cfl(X) —+ Cf‘(X) establishes an isometric isomorphism between (Cfl ( X Dr: X and Cf‘ (X) and it commutes with the boundary 79 operators. Thus the 61 homology group of 1r1X coincides as topological vector spaces with the homology of the complex —> C§‘(X) —s 0f1(X) —s 051(X) —s 0. It remains for us to prove that the isomorphism constructed between Hf‘(1r1X) and Hf‘ (X) is an isometry. Let H - ”1 denote the canonical seminorm in H51 («1X ) and H - HiI the seminorm in H51 (X). By definition of the canonical seminorm, we have M - Ill 2 H - Hf, so that it suffices for us to prove that H - “1 _<_ H - “‘1’. Since the canonical seminorm is achieved by the bar resolution, it suffices to construct a «1X -morphism of the resolution (6.1.1) into the bar resolution of 1r1X consisting of maps of norm 3 1. Let a: An —> X be a singular n-simplex the first vertex of which is in X 0, where An 2 [v0,--- ,vn]. We define a map fn: Cf,‘(X) ——) C,‘,‘(1r1X) by fn(o) = g0[g1|ggl - - - lg"), where g,- E 7r1X such that 0(1),) 2 g.- - --g0X0. It is easy to see that fn commutes with the boundary operators and so it determines a 1r1X -morphism of the resolutions —-—s C§‘(X) ——> Cf‘(X) -—s 051(X) —+R——>O 1 112 11 110 (.... 1 ——> 051(11'1X) ——> Cf‘(1r1X) ——+ Cé‘(1r1X) ————> R ———> 0 extending idR. Also, from definition, it is clear that [I f...” S 1 and so we have M - ”1 3 ll - Hi. This finishes the proof. [:1 Corollary 6.2. Let (1: X1 —> X2 be a continuous map such that the induced ho- momorphism 01.: 171(X1) —> 1r1(X2) is a surjection with an amenable kernel. Then for every n 2 O the homomorphism Hn(oz): H,‘,‘(X1) —> Hf,‘(X2) is an isometric isomorphism for the norm || ~ ”1. Proof. This follows from Theorem 4.9 and Theorem 6.1. C] Now let us define relative {31 homology of spaces. 80 Definition 6.2. Let go: Y —+ X be a continuous map of spaces. The mapping cone and the mapping cylinder of the chain complexes induced by (,0 are defined as follows: C.CO(Y3>X)—so (6.2) —3>ECg(Y ——> X)—d—) EC1(Y —) X)d—> EC0(Y —> X)—> —>.O (6.3) Definition 6.3. The n-th homology of the complex in (6.2) is called the n-th relative 6’1 homology of X modulo Y and is denoted by Hf,‘ (Y 3:) X). We denote the n-th homology of the complex in {6.3) by H£1(EC(Y 3) X). We define the norm II ' “1 On EC.(Y 3) X) by the formula |I(b,x,a)||1= ||b||1+||$||1+||a||1 and similarly on C...(Y 3) X). As is well known, the norm [l - II on M "(Y i) X) in Definition 3.1 is the dual of the 61 norm || - “1 on C...(Y 3+ X). Then, by applying Hahn-Banach theorem, we conclude that the seminorms I] - II on H*(Y 3) X) and H - “1 on Hf1(Y 3) X) are also duaL 81 For every 1.) 2 0, we define a norm || - ||1(w) on Cn(Y 3) X) by the formula “(is a)||1(w) = |I$|l1+(1+w)||al|1- There is the corresponding seminorm H - ||1(w) in Hf‘ (Y :5) X). We define these norms in Hf‘(Y 3) X) for all w 6 [0,00] by passing to the limits. Remark that the seminorms ll . “(6) in H‘O’ is X) and n . ||1(w) in H51 (Y is X) are also dual for every (1) E [0, 00]. Theorem 6.3. Let (p: Y —) X be a continuous map. Then the natural inclusion map pn: Cf,‘ (X) —+ ECn(Y 31) X) induces an isometric isomorphism Hn(p): Hf,l (X) —> Hf,‘(EC(Y 3+ X)) for the norm || - [[1 Proof. We can prove this as the same way as we proved Theorem 5.3. C] Note that there is an exact sequence 0 —s Cf,‘(Y) —s E0,,(Y is X) —s 0,,(Y is X) —s 0 and so there is an induced long exact sequence —s H§1(Y) —s Hf,1(X)—s Hf,‘(Y is X) —s Hfi1_1(Y) —s Theorem 6.4. Let to: Y —+ X be a continuous map and (0.:1r1Y —> rrlX be the induced homomorphism. Then the groups H£1(1r1Y g, 1r1X) and Hf,‘ (Y 3) X) are isometrically isomorphic. This isomorphism carries the canoni- cal seminorm || - ||1(w) in Hf,‘(7r1Y fl) 1r1X) to the seminorm in H51 (Y 3) X) for every w E [0, 00]. Proof. Let G and A denote the groups 1r1X and 1r1Y respectively. Recall that, from Theorem 5.4, the canonical seminorm on H,‘,I(A 31) G) is in- duced by the complex Cn(A g) C) = Cf,‘(G)G ® CfiLl(A)A. 82 Let rrl : X —+ X and m: y —> Y be the universal coverings. As we saw in Theorem 6.1, we can identify Cf,1(XX)@Cf,‘_1Y—(C,‘Q(X))(mx@ (05,1_( ))my. Then, as the same way as we proved Theorem 3.3, we can show that there are chain maps Ch“; )GC$£11- 1( _>(C£1(X )(X$1r (C€11( ))WIY Wu such that \Ilnén is chain homotopic to id and they have the norms “(1),,“ S 1 and |]\IJ,,|| S 1 for the norm || - [II(w) for every (0 E [0, 00]. Cl Corollary 6.5. Let (p: Y —> X be a continuous map of spaces such that the funda- mental group 7r1Y is amenable. Then H,‘,I(X) and Hf,‘(Y 3+ X) are isometrically isomorphic for the norm [l - ”1 for every n 2 2. Furthermore, the norms [I - ||1(w) in the group H,‘,1(Y 5 X) are equal for all 6 [0,00] and for all n _>_ 2. Proof. Let cp,: 1r1Y ——) 7r1X be the homomorphism induced by (p: Y —-> X. We have the following sequence of isometrically isomorphic groups H,‘,‘ (Y is X) a»: Hf,‘ (11'1Y i‘s 1r1X) by Theorem 6.4 E“ Hf,‘ (1r1X) by Corollary 5.10 2’ Hf,‘ (X) by Theorem 6.1. The second part also follows from Theorem 6.4 and Corollary 5.10. C] Let Y and Y C X be a pair of spaces and let go: Y —) X be a natural inclusion map. Then the injective homomorphism in: C,‘,‘ (Y) ‘—) Cf,‘ (X) induces an exact sequence 0 —s 05,10”) gs Cf,‘ (X) —> Cf,‘(X)/Cf,1(Y) —> O. (6.4) 83 We denote Cf,‘ (X) /C,€1(Y) by Cf,‘ (X, Y). Remark that the induced sequence ... —s 0§1(X,Y) —s 051(X, Y) —s 0f1 (X, Y) —s 0 (6.5) is a complex. Definition 6.4. The n-th homology of the complex in (6.5) is denoted by Hf,‘ (X, Y). The exact sequence (6.4) induces a long exact sequence —s Hf,‘+,(X,Y) -—) H§1(Y) —s Hf,‘(X) —s H§1(X,Y) —s H,‘,1_,(Y) —s As in Hf‘ (G, A), for every 6 2 0, we define a norm ||c]]1(0) of c E Cf1(X) by putting ‘ ||C||1(9) = ||C||1 +9||3€H1- Then, by using the surjective homomorphism p...: C51 (X) —> Cf1(X,Y), we define the norm ||5||1(6) of 0' E Cfl(X,Y) by setting ||5|l1(9) = inf{ llCll1(9) l C E 1).-1(5) }- With this norm || - “1(0) on C51 (X, Y), we have the corresponding seminorm I] - “1(0) on Hf‘ (X, Y). Finally we define this norm on Hf‘(X, Y) for all 0 E [0,00] by passing to the limits. Theorem 6.6. Let go: Y E) X be an inclusion map of spaces Y C X. Then the groups H£1(Y 3+ X) and H£1(X, Y) are isomorphic and the norm [l - ||1(w) in Hf,‘ (Y 3) X) and the norm I] - ”1(0) in Hf,‘(X,Y) are equivalent for every w = 9 E [0,00]. Proof. We can prove this as the same way as we proved Theorem 5.12. D 84 CHAPTER 7 Locally finite 21 homology of spaces Throughout this chapter, if (p: X’ H X is a natural inclusion map for a pair of spaces X’ and X’ c X, then we denote Hf1(X' is X) by Hf1(X' L—s X). Now we let I 2 {K, | K, is a compact subspace of X, j = 0,1,2,-~}. We define a relation 3 on the set I by K J- S K,- if and only if K J- C K,. It is easy to see that (I, 3) forms a directed quasi-ordered set. Then, whenever K j, K,- E I satisfy K ,- S K,, there is an injective homomorphism pn: Cf,‘(X - K,) ——> Cf,‘(X — Kj) induced by the inclusion X — K, C X — Kj. Thus there is a chain map 6;: Cf,‘()( X)@0f,1( (—X K) )—s Cf,‘(X X$)Cf,l( (X X,) defined by the formula an(x, a) = (x, pna) = (x, a). This induces a canonical homo- morphism H,,(e;l): Hf1((X — K,)1—+ X) —s Hf1((X — Kj) c—s X). Thus the groups H51 ((X — K) L—> X) form an inverse system indexed by the compact subspaces K C X. Note that [la'llz 1 for the norm || ||1(w) and so we have ||H,,(a})]| g 1 for the norm || - |]1(w) for every 6) E [0, oo]. 85 Definition 7.1. Let I be the index set as above. For every n 2 0 we define the n-th locally finite 61 homology of X, denoted by Hfj"(X), as the inverse limit of an inverse system {Hfil((X — K,) E) X),H,,(a;)}, i.e., H:° gt_n Hf1((X—K')c—sX) KEI K’EI’ called the injection of the system into the subsystem, induced by an index inclusion I’ 1—) I. C). The inverse limit Hf°(X) is a subgroup of a product of the relative 81 homology groups Hf1((X — K,) L—) X), i.e., Hf°(X) = hm Hf1((X—K,-) LiX) KJ'EI z {([st e HH51((X — Kj)1—) X) | [31,] = H,,(n;1)(e,), whenever K, s K,} .7 From now on, we always represent an element [x] of Hf°(X) as an element ([x,]) of a subgroup of the product H]. Hf‘((X — K,) c-—) X). Definition 7.2. We define the seminorm I] 1 IIs in Hf°(X) by putting “(fills = Supllllelll f07‘ every l$l = ([331]) E H3°(X)- J 86 Also, for every w 2 0, we define the norms || 1 ”((60) in Hf°(X) by |I[cv]||1(w) = lijmlllellldw) for every [£6] = ([3311) 6 H3°(X)1 Then we define these norms in HS°(X) for all to E [0,00] by passing to the limits. As explained above, since the map Hn(oz;-) satisfies ||H,,(a})|| S 1 for the norm I] - ||1(w) for every (.0 E [0,00], the limit exists. Also the norm ll - ”.9 may be infinite and H - l|z(w1) S H ' |Iz(w2) for an S 11121 Proposition 7.1. Let the index sets I and I’ be as in Remark 7.1 B). Then the isomorphism QP: i-iI—an—I Hf1((X—K) H X) —> Liliana, Hf‘((X—K’) H X) induced by the inclusion I’ H I is isometric for the norm I] - H, and the norms || - ||)(w) for allw E [0,00]. Proof. Let [x] = ([xK]) E Hf‘ (X) C HKEZ Hf1((X — K) H X). We denote @([x]) by ([xKr]) E HKIEIIH51((X—KI) H X). Since I’ is cofinal in I, there exists K’ E I’ for each K E I such that K S K’ and so there is a homomorphism H.(ozK): Hf1((X — K’) H X) —) Hf1((X — K) H X) induced by an inclusion X — K’ C X — K. Note that, since [x K] = H...(ozK)[xK:] from definition of the inverse limit, we have ||[xK]||1 = ||H.(ozK)[xK)]||1 S |I[th]||1. Thus the norm of every coordinate [xx] of [x] is bounded by the norm of some coordinate of @([x]) so that we have sup K61 H [x K] “1 S supxsezs ||[xK)]||1. Hence “45H 2 1 for the norm [I 1 ]|,. On the other hand, since the supremum taken over I is not less than the one taken over the subset I’ C I, the map 45 has the norm [[45]] S 1 for the norm || - “3. Thus the map 45 is an isometry for the norm ”1”,. Similarly, for the norm || 1 ||)(w), it is easy to see that Lug} Illixllldw) = 113161;}, lll$K1l||1(W) for all w 6 [0,00]- Hence 45 is also an isometry for the norm || - ”((10) for every w E [0, 00]. Cl 87 Corollary 7.2. For [x] E Hf°(X), we have ”[33le S ||l$l||z(w) f0?“ every £1) E [(1)00]- Proof. We prove that ||[x]||, = ||[x]||)(0). By Proposition 7.1, we consider the cofinal index set I’ C I in Remark 7.1.B. Then there is a sequence of groups induced by inclusions —s Hf1((X — K2) H X) —s Hf1((X — K1) 1—s X) —s Hf1((X — K0) H X) —s 0. Let [x] = ([xJ]) E Hfo(X) C H]. Hf‘((X — KJ) H X) be given. Then we have H,(ai+l)[xJ+1] = [le and SO Hllelll = llHn(ai+1)l$j+1ll|1 S ||l$j+1ll|11 J .7 Since [|[xJ-]|]1 = ]][xJ-]]|1(0), we have an increasing sequence of norms o s Illxolll1(0) s II(nHMO) s s |I[Hslll1(0) s (Hes-him) s This shows that limJ- ||[xJ-]||1(0) if the sequence is bounded above, sup |l[$sl|l1 = J limJ- |][xJ1]||1(0) = 00 if the sequence is unbounded. Hence we have ()1th = sup,- lllxsllll =11m. (Hes-111(0) = Ilieilhm) s lllxlllz(w)- D Definition 7.3. For an orientable n-manifold M, we define its simplicial volume as the norm |][x]||, of [x] = ([xK]) E H,‘,’°(M) and denoted it by “M”, where [xK] is the homology class of H£1((M — K) H Ad) for every compact K C M. When M is not orientable we pass to the double covering 117 and set ”M“ = éllfill. Remark 7 .2. Note that if M is a closed orientable n-manifold, its simplicial volume is the norm I] 1 [[1 of its fundamental class in Hf,‘(M). The locally finite 61 homology Hf°(X) is corresponding to the homology T—T.(X) in [7] which is defined with the locally finite cycles 0 = ZZer, such that each 88 compact subset of X intersects only finitely many (images of) simplices 0,. Note that the €1-norm ”0” = 2:, |r,-| may be infinite. Also Gromov defined the simplicial volume of an open manifold M by the norm || - “1 of the fundamental class of 71-.(M). It seems that our simplicial volume of an open manifold is equivalent to Gromov’s simplicial volume but we leave it as an open question. Definition 7.4. A pair of spaces (X ,Y) is said to be perfect if the inclusion homo- morphism 1r1Y —> 1r1X is injective. Remark 7 .3. As it is well known, we can always construct a perfect pair of spaces from a given pair of spaces (X, Y) by killing the kernel of the inclusion homomorphism p: 1r1Y —> 7r1X. Namely, for each [w] E ker(p), we represent it by a loop w in Y which is null homotopic in X. To X we attach a two dimensional disk along the loop w as boundary. Attaching disks in this way we can kill the entire kernel of the inclusion homomorphism p: 7r1Y —> 71'1X . Let X and Y be the resulting spaces of attaching 2-dimensional disks to X and Y, respectively, to kill the entire kernel of p. Then it is clear that this pair of spaces (X, Y) is perfect. Note that the fundamental groups 1r1X and «IX are isomorphic and so Hfl (X) and H fl (X ) are isometrically isomorphic. Also note that the inclusion homomorphism 7r1Y —> fllYis surjective and its kernel is equal to the kernel of {p: 1r1Y —-> 7r1X} so that the groups 71317 and p(rrlY) are isomorphic. Definition 7.5. By the perfect pair of spaces induced from a given pair of spaces (X ,Y) we understand a pair of spaces (X, Y) constructed from (X ,Y) by attaching disks to kill the entire kernel of inclusion homomorphism 7r1Y —-> 7r1X. Proposition 7 .3. Let X and Y be the connected countable cellular spaces. Let (X, Y) be the perfect pair induced from (X ,Y). If Y is amenable, then the groups H51 (Y H 89 X) and H5107 H X) are isometrically isomorphic for the norms || 1 ||1(w) for every w E [0, 00]. ~ . Proof. Recall that the group 1r1(Y) is isomorphic with the image of the homomor- ~ phism 7r1Y —> 7r1X. So the group 7r1(Y) is amenable and so the groups Hf‘ (Y) is trivial. Also remark that there is a commutative diagram H£1(Y) ——1 H51 (X) 111% H£1 H:L. H£11(X) [Ham Ham] Ham] tin-1(7)] H.._1(n)[ H5107) —s H51(X) iii—s Hf,1(YHX,) —s H§L,(Y) ——s H,‘,1_,(X) Note that the map 111(7) is surjective and the map H...(a) is an isometric isomor- phism for the norm H 1 ||1(w) for every 1.1 E [0, 00]. So the map Hum) is surjective by Five Lemma. From Corollary 6.5, the map Hn(q) is an isometric isomorphism for the norm || 1 [II(w) for every w E [0, 00]. Thus the composite Hn(q)Hn(a) is also an isometric isomorphism. Then, since |IH,,(p)|| S 1, we have 1= llHn(Q)Hn(a)|| = llHn(fi)Hn(p)|| S llHn(B)llllHn(p)|l S llHn(B)|| for the norms ||1]|1(w) for every w E [0, 00]. Hence we have 1 S ||H,,(fl)|| for the norm ||1||1(w) for every a) E [0,00]. Since it is clear that we have ||H,,(fi)]| S 1 for the norm || 1 ]|1(w) for every to E [0,00], the map Haw) is a surjective isometry. Since an isometry is injective, the map Hn(fl) is an isometric isomorphism for the norm ||1||1(w) for every w E [0, 00]. [:1 Corollary 7.4. Let X and Y be the connected countable cellular spaces. If Y is amenable subset of X, then the groups Hf,‘ (X) and H£1(Y H X) are isometrically isomorphic for the norm || 1 ||1(w) for every w E [0,00] . Proof. Let (X, ) be the perfect pair induced from (X, Y). Then the following groups 90 are isometrically isomorphic for the norms I] 1 ||(w) for all w E [0,00]: Hf,‘ (Y H X) 9:” H,’,‘(Y H X) by Proposition 7.3 ’.-_‘-’ Hf,1 (X) because 7r1Y is amenable a“ Hf,1 (X) because 1r1X and 1r1X are isomorphic. 1:] Definition 7.6. A subset L of X is said to be large if the complement X — L is relatively compact, i.e., the closure of X — L, denoted by X_-_——L, is compact subset of X. We call the space X amenable at infinity if every large set L C X contains another large set, L’ C L, such that L’ is an amenable subset of L. Theorem 7.5 (Equivalence theorem). Let X be a connected countable cellular space. If X is amenable at infinity, then for every n 2 2 the norms on Hf,”(X) eatery ll - (the) = II - n. for every n e [Gee]. Proof. Let n 2 2 and let X 2 L0 3 L1 3 L2 3 be a sequence of large open subsets of X such that LJ+1 is an amenable subset of LJ- for j _>_ 0. Note that X —LJ- = X — LJ- is a closed compact subset of X. We set K J- = X —LJ- so that LJ- 2 X — K J-. Then {K J} is an increasing sequence of compact subspaces of X and so there is a sequence of homomorphisms induced by inclusions: —s Hf,‘((X — K3) 1—s X) —s Hf,‘((X — K2) H X) —s Hfi1((X — K1) H X) —s 0. By definition Hg"(X) = lir_njH,€1((X — KJ) H X) and for [x] E Hg"(X) we have Illrllls = (Meme) 3 (Menace) for every n e [0.00). We claim that the norms I] [x] ”((7.0) are equal for every w E [0, 00]. Then, especially, the norms ||[x]||)(w) are equal to |I[x]|])(0) for every w E [0,00] and so the theorem will be proved. 91 For eachj 2 1, let pJ-z 1r1LJ+1 —> 1r1LJ- and rJ-z 1r1LJ- —-> 1r1X be homomorphisms induced by inclusions. Then r1 0 pl 0 1 - 1 o pJ = Tj+1 and TJ o pJ- = rJ-+1. Since T1(7I'1L1) is an amenable subgroup of 1r1X and pJ (1r1LJ-+1) is an amenable subgroup of 1r1LJ1, the group 7'1 0 pl 0 o pJ- (1r1LJ-H) = rJ-+1 (1r1LJ-+1) is an amenable subgroup of 1r1X. Thus every large set LJ- = X — K J- is an amenable subset of X. Then by Corollary 6.5 and Corollary 7.4, for every w E [0,00], the norms |I[xJ-]]|1(w) of [xJ] E HT€1(LJ- H X) are equal. Hence for every w E [0,00] the norms ||[x]||)(w) = limJ~ || [xJ]||1(w) are also equal. This proves our claim and so the proof is finished. Cl In the following Vanishing and Finiteness theorems on Hfl (X), we are concerned with the amenable covering of X and so we assume the spaces and their coverings satisfy the conditions for amenable covering in Section 1.3.2. First we state Theorem on Double complexes, which we will use technically for the proof of Vanishing theorem. Theorem 7.6 (Theorem on Double Complexes). Let (Cm) be a first quadrant double complex with differentials (9Q: CM —+ Cp,q_1 and 6p: Cm —+ Cp_1,q, and let T n be its total complex. Also, for each p 2 0, let Mp denote the cokernel of the differential 61: 012.1 ——> Cp,0- Then M. together with the differential 6 is a subcomplex of the total complex T.. Furthermore, if the complexes (CWHB) are exact, then the inclusion M, —> T, induces a homology isomorphism H..(T,.) —-> H...(M_). This Theorem on double complexes is a special case of Theorem 4.8.1 of Chapter I of Algebraic Topology and Sheaf Theory by R.Godement. Its standard proof is based on the spectral sequences of a double complex. However it is also easy to prove directly, using a diagram chase. 92 Theorem 7.7 (Vanishing theorem on Hfl(X)). Let U = {UJ} be an amenable covering of X. If every point of X is contained in at most m subsets UJ- for some m 21,2,111, then |][x]|]1 = 0 for every [x] E Hf,‘ (X) and for every n 2 m. Proof. First, we show that the theorem reduces to the case that the fundamental groups of all components of elements of the covering and their finite intersections are amenable. Let U be some component. As in Remark 7.3, we attach to X the disks to kill the entire kernel of 7r1U —) 1r1X. Then we get a new space X U {attached disks} and its new covering 11’ by including the attached disks. Since 11’ has the same nerve of U , we may consider the space X with covering L1 as the new space with the covering Ll’. So we can reduce the theorem to the case that the group 71'1UJ- is amenable for every U J- E U. In the proof we consider this case only. Let N be the nerve of the covering a. For every 0 E N, we denote by |o| the intersection of elements of the covering corresponding to its vertices. For p, q 2 O, we set 0,,(,N 0:1): G9 051( ([0]) 06M, where N,D is the set of p—dimensional simplices of the nerve N. Taking the direct sum of the complexes 1e 051 (lei) 2, 051 (lei) is 051(lel) e H —> 0 over a E Np, we get a complex for each p Z O -—) 0,, (N, 051) is 0,, (N,0f1) is 0,, (N,051) is 0,,(N) —s 0, where C,D (N) are the real simplicial chain groups of the nerve N. If the simplex o E NJD has the vertices v0 < v1 < 1 1 - < up, we denote by 8,-(0), the (p — 1)-simplex {'00,- 11 ,0,,--- ,v,,} for i = 0,111 ,p. Then the inclusions C§1(|o|) H C51 (Ia-0|) induce a chain map Cp (N, C;‘) —> CV1 (N, C:‘) for every p 2 1. 93, We set 6 = f:O(—1)‘6,1 which is a chain map. Similarly, together with the inclusions C§1(]o|) H C51 (X), we define a chain map —s00()v, 051) —i—s 06(N,0f1) is 00(N,051) is 0001/) ———) o n] n] n] n] —s0§1(X) J—s 0f1(X) is 051(X) —"—s R -——)0. Then we have the following commutative diagram (7.7.1) —s02(N,0§1) is 02(N,0f1) i—s 02(N,051) —"—s 02w) —s o a] a] a] a ——)Cl(N,C§‘) is 01(N,Cf1) ——a—> 01(N,051) —’—’—s 01W) ———s 0 11 11 11 11 _scO(N,0§1) is 00(N,0f1) —6—> 00(N,051) —"—s 00w) ——s o 11 11 1‘1 11 —s0§1(X) is 0f1(X) ——a—> 051(X) —”—s R ——s0. Note that both rows and columns in the diagram (7.7.1) are complexes and the family (Cp(N, C(51))p q>0 together with the differentials 6 and 6 is a double complex. Now, we consider the total complex T defined by T,, = {B 0,,(N,0§1), p+q=n where the differential dnl: Tn —+ Tn_1 is given by the formula dnle(N, C51) = 8+ 6. Then the diagram (7.7.1) defines maps T... —) C? (X) and T, —> C... (N), where C51 (X) and C. (N) are the lowest row and the right column of the diagram (7.7.1) without the group R. Note that the homology of the complex T... is the 21-homology Hf1(T). Remark that for every 0 E N the group 1r1(|o|) is amenable and so Hfl(|o]) = 0. Hence the rows in the diagram (7.7.1) except the lowest one are exact. Thus we have an isomorphism Hf‘ (T) —> Hf1(X ) by Theorem 7.6. Now we claim that the columns in the diagram (7.7.1) except the right one are also exact. To prove our claim we give an alternative description of Cp (N, C51). 94 Let 5,, be the set of singular q-simplices of the space X. For every c E S, we let Nc be the subcomplex of N consisting of the simplices o E N such that c C [0], i.e., for 0: Ag —> X, the image C(Aq) C ]o|. Then the complex CJD (N, C51) = GeeNp C51 (lol) has one basis element for every pair (a, c) E Np x 5,, such that c C [0]. We consider the complex l—lees, Cp(NC). We shall show that the complex $06M» C51 (lol) is isomorphic to a subcomplex of Bees, CP(NC). For every 0,- E Np, we can write an element x,- of 051 ([0,-l) in the form 00 33; = 2731625 6 051(laills i=1 where 0,-J-z Aq —-) [0,] C X and r,J- E R. Let t =®w€®qmn 0561)]? By reordering o,- if necessary, we may assume that only the first k coordinates x1,111 ,xk are nonzero. Then, since 0,1 E NJ) and c,-J-(Aq) C [0,-I, we have r,J-o1,- E C,,(NC,J). If 0,-J = 0,)J-l, by arranging them nicely, we can define a map , z I“... 69 0,1(iel) —> H CAN GEN? 665,, It is clear that the map PM is well-defined and injective. Also note that (ZMW=iimm j:1i1: i=1 j=1 00 We denote by flees, BP(NC) the image of PM. It is clear that the sequence WeESq CESq CESq is a complex, where the differential 6’ : [Lesa BJ,(NC) —> HCESQ Bp_1(Nc) is induced from the boundary operator Cp(NC) —> Cp_.1(Nc). Also the complex 6909,? C51 ([0]) is isomorphic to the complex flees, BJ,(NC). 95 Moreover, an examination of the definition of the maps 6 and u shows that the map 6: Cp(N, C51) —+ Cp_1(N,C§1) is equal to the map noes, Bp(Nc) ——> Flees, Bp_1(NC), and similarly for u. Thus the columns except the right in the diagram (7.7.1): is 0201/, 051) is 01(N,0;1) is 00(N,0;1) is C“(X) —s o q is isomorphic to the sequence of complex in (7.7.2). Note that the subcomplex Nc is acyclic so that H..(Nc) = 0. So the sequence (7.7.2) is exact and so our claim is proved. Then, by Theorem 7.6, we have an isomorphism Hf‘ (T) —> H. (N) Note that the homology of the complex C. (N) is actually the homology of the simplicial scheme of N which coincides with H..(|N I), where [N] is the geometric realization of the nerve. Since every point of X is contained in at most m elements UJ1 of the covering U of X, Hn(|N]) = O for every n 2 m. Thus, for every n 2 m, we have H§‘(X) = H.310") = H,,(N) = 0 and hence [|[x]||1 = 0 for every [x] E Hf,‘ (X). So the proof is finished. [I Corollary 7.8 (Vanishing theorem on Hf°(X)). Let L! = {UJ} be an amenable covering of X. If every point of X is contained in at most m subsets UJ, for some m 21,2,111, then ||[x]||s = 0 for every [x] E Hf(X) and for every n 2 m. Proof. As in Theorem 7.7, we can reduce the theorem to the case that the group 7r1UJ- is amenable for every UJ- E U. By Corollary 7.4, for every UJ- E Ll, the groups H? (X) and Hf‘ (X, UJ) are isometrically isomorphic for the norm || 1 ”1. Then by Theorem 7.7, the group Hf,‘ (X) and so the group Hf,‘ (X, UJ) are trivial for every n 2 m. 96 We consider Ll-small singular simplices. Let n 2 m, and let I = {K,- | K,- is a compact subspace of X, i = 0,1,2,---}. For given [x] = ([x,]) E H°°(X ) C HK E1H ,i‘((—X K.) H X), we represent each [x,-] E H,‘,1((X — K,) H X) by a cycle (z,,a,-) E C£‘(X)®Cf,‘_1(X — K,). Note that 8:5,- : —a,-. We denote by zf the linear combination of n-singular simplex o in 2,- such that Ono is entirely contained in UJ. Then we can write 2,1 as a sum of j z . 1) i.e., z,- 2 J23. Note that |]z,||1 S 2: stjlll and also note that (2?, —8z,j) E C,‘,1(X)$C,’,‘_1(UJ1) is a cycle. Thus [23, —(9z,j] E Hf,‘(UJ- H X) = 0 and so “[23, —sz]||1 = 0. 1, Now we have lllwsllll < il( )iil < 2M2.- —ee.1)iis, and so |][x,-]||1 = 0. Hence “MM, = sup K, ]I[xil|l1 = 0 and so the proof is finished. Cl Theorem 7.9 (Finiteness theorem on Hf°(X)). Let U = {UJ} be an amenable open covering of X such that UJ- is relatively compact. If there is a large open set every point of which is contained in at most m subsets UJ, for some m = 1,2,-11, then the norm ]|[x]||, of [x] E Hf(X) is finite for every n 2 177.. Proof. Let n 2 m. Let L be a large open set every point of which is contained in at most m subsets UJ. We set K0 = _X_—_I_s. Then K0 is a compact subset of X. By reordering the indices j if necessary, we may set for some index k0 K0: UT]; and L: UU,. jSko j>ko As before we may reduce the theorem to the case that the group 1r1UJ~ is amenable. We consider U-small singular simplices. Note that the set {UJ- E U | j > k0} is an 97 amenable covering of L. Hence, by Corollary 7.8, we have ||[y]||, = 0 of [y] E H? (L) for every n 2 m. We define K,- by K,- = K,_1 U m inductively on i 2 1. Then there is an increasing sequence of compact subsets K0 C K1 C K2 C such that U:O(K,)° = X. So, by Remark 7. 1. C and Proposition '7. 1, we can define H°°( (=X) EEK H,‘,‘(( X— K,) H X). Let [x] = ([x,]) E H§°(X) C UK,H1€1((X — K,) H X) be given. We represent [x,] by a cycle (2,,c,) E C,’,1(X)$C,’,‘_1(X — K,). Note that 62,- = —c,-. Let a, be a linear combination of n-singular simplices in 2, such that whose images are entirely contained in L = U UJ- and b, be the rest of 2,- after choosing a,. Then 2, is written j>ko as the form 2,- : a, + b,. Note that a, E Cf,‘ (L) such that 8a,- E Cf; 1(L— K,.) Thus (a,, —8a,-) E Cf,‘(L)®Cf,‘_1(L—K,) is a cycle. By applying to [a,, —Ba,~] E Hf,‘((L—K,) H L) the same argument on [2,, —(92,-] E Hf,1((X— K,) H X) and [2,] - ,—82,’] E H‘1(UJ- H X) in the proof of Corollary 7.8, we have ]][a,, —8a,~]|]1 = 0. Also note that b, E Cf,1(Ko) and 8b,- E C,‘,‘_1(K0 0 (X — K,)). Since the spaces K0 0 (X — K,) is empty for all i 2 0, we have 6b,- = 0. Thus b,- E C,‘,‘(K0) is in fact an n-cycle on K0 for every i. So we have [b,-] E Hf,‘ (K0) and ]|[b,]|]1 < 00 for every i. Recall that for every i Z 1 the homomorphism H..(n:,):Hf,1((X — K,) :s X) —s H,‘,‘((X — K0) 1» X), induced from the inclusion 013: Cf,‘ (X) Q Cf,‘_1(X — K,) —-> Cf,1(X) ® Cf,‘_1(X -— K0), satisfies Hn(afi,)[2,, —82,] = [20, ~620]. So we have [2,, —82,] = [20, —820] in H,‘,I((X — K0) H X). Thus there IS an element (2’, 0’) )E Cf1 (X) @CfiL 1( (X— K0) such that (2,, —82,) — (20, —820) = (a,- + b,, —8a,~ — 8b,) —— (a0 + b0, —8a0 — abo) = (a, — a0 + b, — b0, —Ba, + 800) = B(z], c1) = (62; + c], —Bc;). 98 As before, we rewrite 2,’- = a] + b; so that a; and b; are the linear combination of singular simplices in 2,’ such that whose images are entirely contained in L and in KO respectively. It is easy to check b,- — b0 2 0b]. This shows that [b,] 2 [b0] and so lllbillll = HlbOlHl S 001 Thus we have lllxllls = SUP “ll‘z‘llli = SUP lllzis _(9211111 = SUP ”[111 + 1),, —6a, — abillll K. K- K; I 8 S SUP Illais “aaillll + SUP lllbz'llll = lllbOlHl < 00- K, Ki Hence the norm |I[x]|], of [x] E H,‘,’°(X) is finite for n 2 m. 1:] Remark that if X is an m—dimensional manifold satisfying the assumption for Finiteness theorem, then the simplicial volume of X is finite. Also if X satisfies the assumption for Vanishing Theorem on Hf°(X), then the simplicial volume of X is zero. 99 BIBLIOGRAPHY 100 BIBLIOGRAPHY [1] G.Bredon, Topology and Geometry, Springer-Verlag, New-York, 1991. [2] R.Benedetti, C.Petronio, Lectures on hyperbolic geometry, Springer-Verlag, New- York, 1991. [3] R.Brooks, Some remarks on bounded cohomology, Ann. Math. Studies, 97, 1981, 53-63. [4] K.S.Brown, Cohomology of Groups, Springer-Verlag, New-York, 1982. [5] M.Greenberg, J .Harper, Algebraic topology-A first course, Addison-Wesley, 1981. [6] R.I.Grigorchuk, Some results on bounded cohomology, London Math.Soc., 202, 1993, 111-163. [7] M.Gromov, Volume and bounded cohomology, Publ. Math. 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