|HIM”NHWWWWWWI POINCARE DUALITY SPACES Thesis for the Degree of Ph. D. MICHIGAN STATE UNEVERSiTY OLIVER COSTICH 1969 TH. E152“. 11111111111111111 1/ 321 3 00705 6983 This is to certify that the thesis entitled "Poincare Duality Spaces" presented by Oliver Lee Costich has been accepted towards fulfillment of the requirements for _P_h_-.Q__ degree in wt ic s /C WW Major professor Date AUQUSt 4. 1969 0-169 2 at. .C ABSTRACT POINCARE DUALITY SPACES by Oliver Costich This thesis is a study of algebraic conditions which will guaranteethatziquotientSpace of a manifold is itself a manifold. Suppose X is a locally compact Hausdorff space and A is a compact, connected subspace of X. Moreover assume that there is a "Poincaré Duality" isomorphism A of the cohomo- logy of X onto the homology of X, A : Hq(X) 4 Hn- (X). A is said to be a divisor of X if the homomorphism A-l n-q n-q Hq(A)-—» Hq(X) ._» H (x)-_4 H (A) is an isomorphism for q % O,n. In Chapter I it is shown that if X is an orientable, compact, polyhedral homology n-manifold, then A is a divi- sor of X for singular homology and cohomology if and only if the quotient X/A is an orientable, compact, polyhedral homology n-manifold. Chapter II demonstrates that if X is an orientable n-dimensional cohomology manifold, then A is a divisor of X for Alexander-Spanier cohomology and Borel-Moore homology if and only if X/A is an orientable, n-dimensional cohomo- logy manifold. In addition, the following generalization of R. L. Wilder's theorem on monotone mappings of manifolds is given. ,.Iheozgm: Let f : X 4 Y be a surjection of a compact, orientable, n-dimensidnal cohomology manifold X to a locally compact Hausdorff space Y. If f-1(y) is a divisor for each y 6 Y, then Y is also an n-dimensional cohomology manifold. POINCARE DUALITY SPACES By /' Oliver(Costich A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1969 ACKNOWLEDGEMENTS The author wishes to eXpress his gratitude to Professor K. W. Kwun for suggesting the problem and for his helpful suggestions and patient guidance during the research. He also wishes to thank the faculty members and graduate students of the Department of Mathematics with whom he has had many stimulating discussions during the investigation. ii TABLE OF CONTENTS ACKNOWLEDGEMENTS O O O O O O O O O O O O O O 0 INTRODUCTION . . . . . . . . . . . . . . . CHAPTER 0. II. PRELIMINARIES . . . . . . . . . . . .., . 1. Notation and Terminology . . . . . . 2. An Algebraic Lemma . . . . . . . . . DUALITY IN POLYHEDRAL HOMOLOGY MANIFOLDS DUALITY IN COHOMOLOGY MANIFOLDS . . . . . l. Homology, Cohomology, and Duality for Cohomology Manifolds . . . . . . . . 2. The Main Theorem . . . . . . . . . . BIBLIOGRAPHY . . . . . . . . . . . . . . iii Page ii CDWNN 1% 1H 26 37 INTRODUCTION Various authors studying mappings of 2-manifolds dis- covered that certain "monotoneity" conditions imposed on the counter-images of points guaranteed that the image was also a 2-manifold. In the study of mappings of higher dimensional manifolds, as might be eXpected, similar conditions were im- posed. .Da£1n111gn: A mapping f : X 4 Y is said to be h. monotone if Hr(f-1(y)) = 0 for all y E Y and r.$ n. However, the identification mapping of the 3-Sphere onto the Space obtained by collapsing a suitable "wild" arc [8, EX 1.1] to a point is n-monotone for all n, but the image is not a manifold. There is a class of spaces for which such "monotoneity" conditions are sufficient. R. L. Wilder has demonstrated that a monotone mapping of a generalized manifold yields a generalized manifold as its image [l6], [17]. In this thesis we prove that conditions closely related to Poincaré duality imposed on counter-images of points give similar results for mappings of generalized manifolds. CHAPTER 0 PRELIMINARIES We): Throughout this dissertation X will denote a con- nected, locally compact Hausdorff space. In fact, all spaces will be locally compact and Hausdorff. We take L to be a principal ideal domain and all homomorphisms to be L-module homomorphisms. If A is a compact subSpace of X, we denote by Y the quotient Space X/A, by c : X a Y the canonical identifica- tion, and by * the point C(A) 6 Y. Notice that the com- pactness of A is essential in order that Y be locally compact. A hnmnlnzx_ihenrx (H,a) consists of a covariant func- tor H from a category of locally compact pairs to the cate- gory of graded L-modules and homomorphisms of degree 0, and a natural transformation a of degree -1 from the functor H on (X,A) to the functor H on (A,¢). The domain of H need not contain all locally compact pairs nor all conti- nuous maps. Indeed, different theories may have different domains. We do require that the domain contain all prOper maps. (g : S 4 T is proper if g-1(K) is compact for all compact K in T.) In addition, we insist that (H,a) satisfy 3 (1) For A d X E (X,A), there is an exact sequence 3 1* 3* a ...4 Hq(A) -—o Hq(X) —-o Hq(X,A) 4 H (A) 4... q-l where 1* = H(i) whenever these homomorphisms are defined. (2) On the full subcategory of connected Spaces Ho behaves as follows: If g : S 4 T is in the domain of H and S and T are connected, then a) If S and T are compact, g* : HO(S) 4 HO(T) is an isomorphism and HO(S) ~ L. b) If S and T are non-compact, g* : HO(S) 4 HO(T) is an isomorphism. Analogously. a.nnhomolagx_theorx (H*,6) consists of a contravariant functor H* from a category of locally compact Hausdorff pairs to the category of graded L-modules, and a natural transformation 6 of degree +1 from the functor H* on A to the functor H* on (X,A) satisfying (1) For A &:X a (X,A), there is an exact sequence ”.9. Hq(X,A) if. Hq(X) l3 Hq(A) 9 Hq+l(X,A) 4 (2) On the full subcategory of connected spaces HO behaves as follows If g : S 4 T is in the domain of H* and S and T are connected, then a) If S and T are compact, g* : HOCT) 4 H°(S) 1.. is an isomorphism and H°(S) u L. H0(T) 4 b) If S and T are non-compact, g* H°(S) is an isomorphism and H°(S) O. The remarks made about the domains of homology theories also apply to cohomology theories. Suppose H;, H* are homology theories and H* is a cohomology theory. By a "cap product" we mean a homomorphism . ' q n . Hm(X) e H (X) .4 Hm_q(X) which is functorial in the following sense Let f : X.4 Y which induces maps f; : H;(X) 4 H;(Y), f : H*(X) 4 H*(Y), and f* : H*(Y) 4 H*(X). Then for a 6 Hé(X) the diagram f*an P A \ H (Y) ,Hm_p(Y) f* f* p an n (x) : Hm_p(X) is commutative for all p, where f*an and on are induced by the "cap product". X is called.acEninnazé_Dnali;¥_£nane of formal dimension n (n-PD) if there is a y e Ham such that yn : HP(X) 4 Hn_p(X) is an isomorphism for all p. The element y 6 H£(X) is called the fundamental class of X. Suppose X is an n-PD and A is a compact, con- 5 nected subset of X. A is called a1di¥isnr of X if (1) Hn(A) ~ Hn(A) = o (2) the homomorphism ¢A : Hq(A) 4 Hn-q(A) defined by qu) 2:. Hq(X) ~ n'qoo if. ammo is an iso- morphism for q f’O, n. When we wish to emphasize the base ring L, we will write H(X,A;L), and "X is an n-PDL". Wm Our next step is to prove an algebraic lemma. .Lsmma_Qil= Let 0 4 A’ i A l A” 4 o and R B’ 4 o . 04B' B be exact and let if : A 4 B be an isomorphism. Suppose further that the diagrams A‘——i‘5A A——l—4A” Oi if and fl Ty are commutative, B’(__k_B Beh—.B” i.e., kofoi = m and jor’loh = t. Then m is an isomorphism if and only if t is an isomorphism. ,Enggf: We will show that e is an isomorphism implies p is. The converse is dual to this. To see p is a mono- morphism, let a’ E A’ such that ¢(a’) = 0. Now f°i(a’) 6 h(B') and so there exists b' E B' such that h(b') = 6 foi(a’). But ¢(b”) = jai(a’) = 0 so b” = 0. Thus foi(a’) = O and since f°i is a monomorphism, a’ = 0. To see that p» is an epimorphism, let b’ 6 B’. Then there is a b e B such that k(b) = b’. Let a' e A” be jor'1(b) and 'S e B be hot'1(a'). Now jor'1(bJB) = a' - jOf-10h0t-l(a') = 0 so there is an a’ 6 A’ for which i(a’) = r'1(hJB). Then ¢(a’) = k°f°i(a’) = kHn'q(N) 1* 11 N .- x, 3 : (N,¢) c: (N,N) c (Lit-N), k : (N,N) c Hn'q(x) (h*v)n J’ 1 lyn a: Hq(X)\ Hq(X) . Combining these results, we obtain (vn)°h* = i*°(an)°k* Composing with 1* ‘we get i*h* = i*°(yn)-l°1*°(an)ok* = 12 oNo(cn)ok*. But i*oh* = j*ok* so oNo(cn)ok* = j*ok*. Hence TN°(“n) = 3* so 3* is an isomorphism. Therefore 3* is an isomorphism for O < q < n. From the long exact sequence of the pair (N,N), we See that Hq(N) = 0 if 0 < q < n-l. Moreover O 4 Hn(N) 1: Hn(N,N) 3 Hn_l(N) 4 O is exact. Since N is an orientable nrphm, Hn_l(N) is a free L-module, so the preceding sequence is Split. Thus L ~ Hn(N,N) ~ Hn(N) e>Hn_l(N). Therefore L ~2Hn_1(N) ~ H°(N) by Poincare duality. This implies N is connected and so N has the homology of an (n-l)-Sphere. To show X/A is orientable, we prove that Hn(X/A) u L. The argument above demonstrates that Hn(N) = 0, so Hn(A) = 0. Hence Hn(X) c: Hn(X/A) is a monomorphism. Because either Hn(X/A) = 0 or Hn(X/A) ~ L [13], Hn(X/A) ~ L. The proof of the reverse implication is obvious from the corollary to the algebraic lemma of the preceding chapter. .Bemazk: The proof of the above theorem also Shows that c*y may be taken as the fundamental class of X/A. .Exgmples: (1) Let T = 3le1 and R c T be the one point union of two circles representing the canonical genera- tors of nl(T>. Then T/R a 32. Thus R is a divisor of T. More generally, if M is a closed, orientable, polyhe- dral manifold, there is a closed subpolyhedron R C'X satis- fying xa-mnun, nnnR=¢ and dimR Scn*1 Thus the sequence 0 d . g 2 d o 4 614 c (xm) 4 c (X34) c, (met) 4 is exact. That is, C*(X;A) is a resolution of d. This is thaw of 0.. Notice that C°(X;d.) is an exact functor of 04 and, therefore, so is C*(X;d). ..h‘ For m a family of supports on X, we put C:(X;d) = I‘m(Cn(X;d)) 1' Cgfin(X;d)) Since C;(X;d) and 2n(X;d) are exact functors of 1, so is C3(X;d). We now make the definition H:(X;d) = Hn(C$(X;4)) From a short exact sequence 0 4 a’ 4 d4 4' 4 O of sheaves on X we obtain a short exact sequence 0 4 c;(X;dj) 4 c;(x;d) 4 c$(x;a.') 4 o of chain complexes and thus a long exact sequence ...4 H£(X;d’) 4 Hgmm) 9. Agnew) 4 Hg+l(X;d) 4... Alternatively, we can define sheaf cohomology via injec- tive resolutions of sheaves. A sheaf .9 on X is m if given a monomorphism i 44 6 of sheaves on X and h : 4 4J, there exists an E : 8 4 .0 such that "hoi = h. That is, the functor HomL (°,J) is exact. From the homological algebra of L- modules, we know that any L-module is a submodule of an injec- l9 tive L-module and that this injective module can be constructed in a canonical way [12]. For a sheaf a. on X, let Max) be the canonical in- jective L-module containing ax' Define J0(X;d) to be the sheaf generated by the presheaf 30(X;d)(U) = MIMX) 1 x 6 U}. Then JO(X;d) is injective and U1Q1X6U}CH{I(AX)|er} provides a monomorphism CO(X;d) 4 JP(X;O.). Composing with the canonical monomorphism a. 4 CO(X;d) gives a monomorphism a 4 30(X;a). Hence every sheaf is a subsheaf of an injective sheaf and thus the standard methods of homological algebra can be applied to sheaves. Define ‘9];(X;a.) = 30(X;0.)/d. and Thom) l°(x,yn(x,a)) where fund) 91(x;9n‘1(x;m). We obtain a resolution J*(X;0.) of d which is a covariant functor of 4.. We refer to t9*(X;d) as the hamm- Min of d. In the usual way, if J* is a resolution of 4., then there is a chain map 1* 4 3*(ng). that is, there is a com- mutative diagram 4—610—‘1411 —9. .. 1 l d—sJo—4a01 ——>- .. n. “mk‘u 0 III an. 1‘“ 'h. I A!‘ u V‘ Cr"! 4. 20 Let 1* = C*(X;d) and let p be a family of supports on X. We then have a chain map P¢(l*) 4 C$(X;d) and hence an induced map P p . H (r¢(.1*)) 4 H¢(X,d). This is a natural isomorphism for all p [5]. Thus we can determine the modules H£(X;d) from. A*(X;d) as well as from C*(X;d). To define relative cohomology, let i : A.c X and m a family of supports on X. If a. is a sheaf on A, there is a sheaf id. on X determined by id(U) = dKUnA) and a continuous map a. 4 id. so that A————>ia- L i L A ;X commutes. If (4,11') is a sheaf on X, (dlAur’) is the sheaf on X given by 41A =dmr-1(A), vr’ = 1r1(4.n1r-1(A)). For 4. on X, we have a homomorphism, i* : C*(ng) 4 in(A;d1A) Of sheaves on X. We introduce the notation Ker i* = CF(X,A;a) c;(x,A;(D F¢(C*(X,A;0.)) H;(X,A;0.) H*(C;(X,A;a)) From these definitions we obtain a short exact sequence 0 4 c;(x,A;a) 4 c;(x;d) 4 anA(A;a1A) 4 o and hence a long exact sequence 21 ...4 H£(X,A;Q) 4 Hg(X;d) 4 HgnA(A;a1A) 4 Hg+1(x,A;d) 4... For this cohomology theory, excision theorems, universal coefficient theorems, Mayer-Vietoris sequences, and many re- sults Similar to those available for "ordinary" cohomology theories are provable [5, Chapter II]. If G is an L-module we also denote the.cnnstant_shea£ (GxX,w), where w : GxX 4 X is the projection, by G. If m and mnA are both paracompactifying, H*(X,A;G) 4 H*(X,A;G) O m AS where the right-hand side is the Alexander-Spanier cohomology module of (X,A) with coefficients in G [5], [13]. If, in addition, X and A are homologically locally connected in the sense of singular homology (HLC), H;(X,B;G) ~ AH;(X,B;G) where the right-hand side is the "classical" singular cohomo- logy module of (X,B) with coefficients in G. To define the homology theory which is "dual" to sheaf cohomology we need some additional objects. A.pnennshaa£ u on X is a covariant functor from the category of Open subsets of X to that of L-modules. A precosheaf is aicnsheaf if the sequence 22mM952uwg4um)4o a is exact for all collections {Ua} of Open sets with U = U , where g = Z} (i - i ) and f = Z31 H a UaUB UBUa a UUc 1 [iUV being the canonical map N(V) 4 u(U) for V c U and d— ‘1) Ian! - ..dl' In. Ipa. l-vd t ..q P. I n...‘ q.“ 22 UaB = UanUBl. Graded and differential cosheaves are defined in a manner analogous to that for sheaves. Let 0 4 L 4 L0 4 L1 4 o be the canonical injective resolution of L, i.e. L0 = I(L), L1 = I(L)/L which is injective since it is divisible and L is a principal ideal domain. We define Morn, (ELL) to be the sheaf generated by the presheaf U 4 Hom (N(U),L). Now define a differential sheaf 3(fl*;L) = na4t(u*,L*) for a differential cosheaf fl*, where L* is the canonical injective resolution of L. As usual, the term in degree n is .Bn(2t*;L) = z mmpmb p+q=n the differential being d’ - d” : en 4.en+1 ‘where d’ is irniuced by the differential Mq 4 Mq+l and (-l)nd' is induced by 21p +1 4 up. Let 1* be a "nice" differential sheaf. Then TCI* is a differential cosheaf with gradation a 'p ‘ (I‘Cl*)p I‘Ca‘. . The differential sheaf h(rCI*;L) will also be denoted by .DU't). Moreover, as above, we let .Bn stand for 19—11. For a sheaf d, on X, we define C¢H§_ (Y;L) f*1/ T f... p . an . C . HC(X,L) ,Hn_p(X,L) is commutative for all p. If U and V are Open subsets of X with V crU, there is a homomorphism Tc(V) 4 PC(U) provided by "exten- sion by zero", i.e. for s E Pc(V), extend s to U by 26 s(x) = O, x e V - 151. This induces a homomorphism jU'V : H6(V) 4 H6(U) which is natural with respect to inclusions. The Borel-Moore homology is frequently equivalent to more familiar homology theories. This is treated in [5]. ikxUfl£EL£L__J&uledlLfllmmuxmh In this section, we prove the main theorems of this thesis. The following may be found in [5]. .2.1 If A is closed in X, then we have the natural isomorphisms. (i) H5(X,A) ~ Hé‘ng(x-A) ~ H5(x-A) (ii) HS(X,A) ~ HE“(X‘A)(x-A). Using these we can prove a .Lemma_2.2: Let A be a closed subset of a locally compact Space X. Then for q > O, (i) Hg(x/A) ~ Hg(x,A) (ii) Hg(X/A) ~ Hg(x,A) are naturally isomorphic. m: (i) Hg(X/A) ~ Hg(X/A,*) ~ H8(X/A - *) ~ Hg(X-A) ~ H8(X,A). The isomorphisms are, successively, obtained from the cohomology sequence of (X/A,*), 2.1(1), relative .1: 27 homeomorphism, and 2.l(i) again. (ii) Similarly, Hg(X/A) ~ Hg(X/A,*) ~ H§“(X/A ' *)(X/A - 4) ~ 330(X'A)(x-A) ~ H:(X,A) from 2.1(ii). Using 2.2(1), we get 2.3 If i : A c-X is a closed subSpace and U = X - A, there is an exact cohomology sequence ...4 ago» 1X3 1130:) ii“. H8(A) 4 Hgflw) 4 .Jfiumm14adi: If W is a nondegenerate, locally compact Space, and w E W, then dimLW = dimL(W¥w). ,Ezggf: In 2.3, take A = {w} and X any Open set in W. .Qfinollany_2‘5: If dimLX < a, then dimL(X/A) < a. .Ezggf: By 2.N, dimL(X/A) = dimL(X/A - *) = dimL(X-A) é dimL(X). We list a theorem about cohomology manifolds. A proof may be found in [2]. [Theorem_2.fi: Let X be a connected n-cmL. Then (T) For every non-empty Open subset U, the homomor- phism jXU : Hg(U) 4 H3(X) is surjective, hence H€(A) = O for every prOper closed subset A of X. (2) x is orientable if and only if HSCX) ~ L. If x 28 is orientable and U is an Open subset, then U is orientable and, if U is moreover connected, , n at n im . new) 4 econ. We also will require a universal coefficient formula relating sheaf cohomology with compact support to Borel-Moore homology with closed support. 2.7 If U is an Open subset of X, there is a sequence 0 4 Ext (H8+1(U),L) 4 Hp(U) 4 Hom (H8(U),L) 4 0 which is natural with respect to inclusions of Open sets, that is, with reSpect to Hp(X) 4 Hp(U) and H6(U) 4 H6(X) [5, p. 18%]. .Lemma_2‘8: Let X be an orientable n-cmL and let U, V be Open, connected subsets of X with U c‘V. Then the homomorphism Hn(V) 4 Hn(U) induced by restriction is an isomorphism. .Enggfl: From 2.6, jXUf and jxv are isomorphisms in dimension n, and since jXU = jXVOjVU’ jVU is also an iso- morphism. The universal coefficient formula produces a com- Imtative diagram o-—4Ext (Hg+1(v),L)-——4Bn(V)-——4Hom (Hg(V),L)-——)O Ext (jVU,l) L if Hom (jVU’1)1 o-——4Ext (H3+1(U),L)-——4Hn(U)-——>Hom (H3(U),L)-——+0 Since dimLX s n, Ext (H8+1(V),L) = Ext(Hg+1(U),L) = o. 29 Moreover, Hom (jVU'l) is an isomorphism since JVU is. Thus Hn(V) 4 Hn(U) is an isomorphism as required. By "divisor", we now mean a divisor with respect to sheaf cohomology and Borel-Moore homology, both having constant coefficients and compact supports. In order to establish that the orientation sheaf of Y is locally constant we need a ‘Lemma_2.9: If X is an orientable n-cmL and A is a compact, connected divisor, then for any Open neighborhood U of * in Y, the homomorphism c*: B8(U) 4 Hg(c'1u) a -..? induced by c : X 4‘Y is a monomorphism for all p and is an isomorphism for p = n. ,Enggf: Since the long exact sequence for cohomology is functorial, the algebraic lemma provides a commutative diagram, p > O 0—9H8(Y)—£:—)H8(X) h*z ;Hg(A>—40 1 1,4 1: ...__,hg(U)—‘L,Hg_k*_.ligm _,... (since A is also a divisor of the n-cmL, c-lU) where h*, k*, and the vertical homomorphisms are induced by inclusions. Since k*or* = h* is an epimorphism, so is k* (for p > 0). Thus c* : H8(U) 4 H8(c-1U) is a monomorphism for p > I, and, because by 2.6 H8(A) = 0, an isomorphism for p = n. Due to the fact that the augmented homology module H8(A) is 30 trivial, c* : H%(U) 4 H%(c-1U) is a monomorphism. For p = 0, any map of connected Spaces induces an isomorphism. Combining the preceding results we are able to show ,,Lemma_2‘lQ: If X is an orientable n-cmL and It is a compact, connected divisor, then the orientation sheaf fih(Y) of Y is locally constant. In fact, it is locally isomor- phic to the constant L-sheaf. Proof This is clear for points of Y other than *. To prove it for *, let U and V be connected neighbor- hoods of * in Y with U c'V. The universal coefficient formula 2.7 provides a commutative diagram (since dimLU, dimLV a n) Hn(V) ’ " sHom (HQ(V),L) r* J] Hom (jVU’l) Hn(U) "’ ;Hom (H8(U),L) where r* is induced by restriction. But in the commutative diagram Hgm) °* ;H3(c'1U) j J _ - VU c 1V,c lU Hgm °* s ng the monomorphisms c* are isomorphisms due to 2.9, and Li”...- V 31 jc-lU,c-1V is an isomorphism which causes Hom (jVU,l) to be an isomorphism. Hence r* is an isomorphism. Define a presheaf F on Y by F(U) = Hn(U’) where U’ is the component of U containing *, and define the homomorphism F(U) 4 F(V) for V C’U by restriction. Since clog is equivalent to local connectedness [l5], and X is clcg, connected neighborhoods of * in Y are cofinal in the neighborhood system of * in Y. It is then easy to see that the sheaf generated by F is isomorphic to the sheaf flh(Y). The above discussion also yields that F is locally constant with stalks isomorphic to L. In order to see that Y is cch, we need that this is equivalent to another condition; 2.11 [5, p. 77] Let X be a locally compact Hausdorff Space, then the following two statements are equivalent. (1) X is clcE (ii) If U and W are Open, relatively compact subSpaces of X with 'U C'W, then Image [jWU : H8(U) 4 H8(W)] is finitely generated for each p. (Here ‘U denotes the closure of U.) «Lemma_2‘12: If X is an orientable n-cmL and A is a compact, connected divisor, then Y is cch. .Bnggf: Since dimLY is finite, we only need prove that 32 Y is clcE. In fact we only need prove that 2.ll(ii) holds where U and W are neighborhoods of * in Y. To see this, consider the following commutative diagram. Bg(u) °* )H8(c-1U) jWU ‘1’ i jc-IW,c-1U Hg(W) °* ng Now 'U c.W implies that c-lU c c-IW, but X is clef so that Image [j _ _ : H8(U) 4 H8(c-1W)] is finitely gener- l l . c W,c U . _ . p ated. Thus we have that Image [Jc-lW’c lU°c* . HC(U) 4 H8(c-1W)] = Image [c*°jWU : H8(U) 4 Hg(c-1W)] is also finite- ly generated. However, since c* : H8(W) 4 H8(c-1W) is a monomorphism by 2.9, we can conclude that Image [ij : H8(U) 4 H8(W)] is finitely generated. Recall that a Space X is,nompletslx_na2asomnact if every Open subset of X is paracompact. This guarantees that closed supports are paracompactifying for every Open subset of X and thus Poincaré duality holds. (Actually, for our purposes it would be sufficient to assume that X and X-A are paracompact). In order to see that Y is an (L-n)-Space we need 2.13 [5, p. 206] The homology sheaf flb(Y;L) .has the stalk over y 6 Y isomorphic to H;(Y,Y-y). [Lemma_2.l&: If A is a compact, connected divisor of \ 33 X, a completely paracompact, orientable n-cmL, then ”h(Y;L) = 0 for q f'n. Since Hg(Y,Y-y) = Hg(X,X-c-l(y)) = O for .Erocf y'f *, 2.13 allows us to consider the single case y = *. Let a 6 Hn(XrA) be the fundamental class of X-A abd Y E Hn(X) be that of X. Consider the "box" diagram Hgf.q(.x-A) JXJX'A 7‘ Hg'qoo 0* (c*a)n The inclusions XrA c X and Y-* c'Y reSpectively in- duce the homomorphisms 1* and k* respectively. The maps cf and c* are all induced by the collapsing map c : X 4‘Y. The right and left faces commute due to the functorial nature of the cap product, and the rear face commutes because Of a prOperty of Poincare duality [5, p. 210]. The tap and bottom faces commute by reason of the fact that the homomorphisms induced by a prOper map on the homology and cohomology modules is functorial. We will demonstrate that the front face also 3h commutes and that consequently k* is an isomorphism. By chasing the diagram we get that (c*v)fl°jY,Y_* c,.,<>vf1°c*°,j1Y’Y_,.l = c*oynojx’x_Aoc* = c*Oi*°anoc* = k*oc*oanoc* = k*°(c*a)n Moreover, the exact sequence 2.3 with A = * and X = Y shows that jY,Y-* is an isomorphism for q‘# 0. But (c*y)n is an isomorphism because A is a divisor, and (c*a)n is an isomorphism because it is the composite, c*°an°c*, of isomorphisms. Thus k* is an isomorphism for q %'n. The long exact homology sequence of the pair (Y,Y-*) then yields that Hq(Y,Y-*) = o for q ,1! n. By 2.13, fld(Y;L)* = O for q f n. We have already noted that VCI(Y;L)y = O for q f’n and y E Y-* so flq(Y;L) is tri- vial for q % n. «Corollazy_2.15: If A is a compact, connected divisor of X, a completely paracompact orientable n-cmL, then Y is an (L-n)-Space. .Eroof: By 2.14, ”h(Y;L) = O for q f n and by 2.10, Nh(X;L) has stalks isomorphic to L so it is torsion-free. Combining our previous results, we obtain the principal results 35 .Iheozem_2.16 If X is a completely paracompact, orientable n-cmL, then A is a compact, connected divisor if and only if Y is an orientable n-cmL. Moreover, in either case, the sequences HC C C 04 (A)4H(X)4H(X/A)4O q q q and 0 4 Hg(X/A) 4 H8(X) 4 Hg(A) 4 0 [T are Split exact for q f 0. [£2o9£: By 2.5, 2.10, and 2.15, Y is a finite-dimen- sional (L-n)-Space satisfying wh(Y;L) is locally constant with stalks isomorphic to L. Thus Y is an n-th. By 2.12, Y is cch and hence Y is an n-cmL. According to 2.6(2), Y is orientable since c* : H§(X) 4 Hg(Y) is an isomorphism. The next theorem.genera1izes a result due to Wilder [16. 17]. «Theorem_2.lz: Let X be a compact, orientable n-cmL and let f : X 4 Y be surjective such that for each y‘e Y, f-l(y) is a connected divisor. Then Y is an orientable n- cmL o lzogi: Let {A1 1 i E I} be the collection of point- inverses of f which are not acyclic. Since H5(X) is finitely generated [15] and since A1 is a divisor, Ham/A1) is simpler than H6(X) in the sense that either rank H5(x/Ai) < rank H500 or the torsion part of H5(X/Ai) 36 is a non-trivial direct summand of the torsion part of H5(X). Since such simplifications cannot be made infinitely many times, I must be finite. Let Z be the space obtained from X by collapsing each A1 to a point. By 2.16 and induction, Z is an orientable n-cmL. The map Z 4'Y induced by f has acyclic point-inverses so by Wilder's monotone mapping theorem [16], Y is an orientable n-cmL. .Example: The examples of chapter 1 are also examples here since an n-pth is an n-cmL. To construct a non-poly- hedral example, let C be a "sin 1/x curve" in a h-cell so that 'C n S3 is an arc, where S3 = 6“. Let h : S3 4 82 be the HOpf map and let A = im [C U S3] in en Uh 82 = 6P2. Then A is a divisor since CPg/A is homeomorphic to S” .Bemazk: If X is completely paracompact and f is prOper, one can drOp the compactness from the hypotheses of theorem 2.17. In this case, choose for y E Y, neighborhoods U and V so that f-1(U) c f-1(V) are connected and rela- tively compact. Then by Poincard Duality and [5. p. 77], im [HS(f-1U) 4 H$(f-1V)] is finitely generated and contains the image of H$(A), for any divisor A in f-1(U), as a direct summand. As in the proof of the theorem, only finitely many such divisors can be non-acyclic, so that U is an n-cmL. [1] [2] [3] [1+] [5] [6] [7] [8] [9] [10] [ll] [12] [13] [1t] [15] [16] BIBLIOGRAPHY 8. AleksandrOV..Qomhinaiorial_1onolosx_lll. Graylock Press (1960). 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