FY“ ELYVGLFJ Yqu‘YS {HY REES SFKCES MB WEEK 3vki1§MFOLES 745i. azeam‘iai: or. th‘ Yam {Dogma of pit. 91 EYE YEEGAN {3%}? EH NW .133 YY Pam; Ree am E973 __..-._- __. . “A A L. J LIBRARY ‘ Michigan State U Universnty _ a q ' mun» -" This is to certify that the thesis entitled ?L.: lNVOLUT|DNS 0N Liz/cs spaces AND OTHER B'HAUtT-OLDS presented by ?a i K Ke e. K q M has been accepted towards fulfillment of the requirements for . Pk‘bqegreein t) M! LEW) "Lo kgméji. K21... U [Mljor professor . : Date 8- 7‘ 73 04639 PL INVOLUTIONS ON LENS SPACES AND OTHER 3-MANIFOLDS BY Paik Kee Kim AN ABSTRACT OF A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1973 ABSTRACT PL INVOLUTIONS ON LENS SPACES AND OTHER 3-MANIFOLDS BY Paik Kee Kim This thesis is to complete the classification problem for sense preserving PL involutions with non-empty fixed point sets on 3-dimensiona1 lens spaces L = L(p,q). The classification problem for PL involutions on the pro- jective 3-space P3 as well as that for PL involutions on P3 # P3 will be settled. The principal results are the following theorems. Theorem 1: If h is an orientation preserving PL involution on L(p,q), p even, which preserves sense and has non-empty fixed point set F, then F is a disjoint union of two simple closed curves. Theorem 2: Up to PL equivalences, there is exactly one orientation preserving PL involution on L(p,q), p even, which preserves sense and has non-empty fixed point set. Corollary 3: Up to PL equivalences, there is exactly one orientation preserving PL involution on P3 with non- empty fixed point set and there is exactly one free involution on P3. Paik Kee Kim Theorem 4: Let h be an orientation preserving PL involution on a lens space L = L(p,q), p even, which pre- serves sense and has non-empty fixed point set . Then there exists a PL equivariant homeomorphism t on L such that t interchanges the two components of Fix(h) if and only if L is symmetric. Corollary 5: Let h be an orientation preserving PL involution on P3 # P3. If Fix(h) = ¢ or Fix(h) is con- nected, h is the obvious involution which interchanges the two P3. If Fix(h) is not connected, Fix(h) is a disjoint union of three simple closed curves and there is exactly one such h, up to PL equivalences. PL INVOLUTIONS ON LENS SPACES AND OTHER 3-MANIFOLDS BY Paik Kee Kim A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1973 .5 ‘ ' 3 2 ..‘ a? "J". ‘4' “G: k (“J-1‘; 3 To my mother and Myung ii ACKNOWLEDGMENTS The author wishes to express his gratitude to Professor K. W. Kwun for suggesting the problem and for his helpful suggestions and guidance during the research. This research.was supported in part by NSF Grant GP 29515X. iii TABLE OF CONTENTS INTRODUCTION.......... ........ . ...... . ............ 1 CHAPTER I. FIXED POINT SETS......................... 3 II. PL INVOLUTIONS ON SOME 3-MANIFOLDS....... 15 BIBLImRAPHYOOOOOOOOOOOOOOIOOOOOOOOOOOO0.0.00.0... 25 iv INTRODUCTION An involution h of a lens space L = L(p,q) is called sense preserving if h induces the identity of H1(L). The purpose of this thesis is to classify the ori- entation preserving PL involutions of L which preserve sense and have non-empty fixed point sets for p even. As results, this thesis will lead a complete classification of the PL involutions on the projective 3-space P3 as well as that of the PL involutions on P3 # P3. We work in the piecewise linear (PL) category. All PL involutions are known on 83 (see Livesay [5,6] and Waldhausen [20]) and on 81 x 82 (see Fremon [3], Kwun [8], Tao [18], and Tollefson [19]). Therefore, in this the- sis we will not consider S3 and S1 x 52 as lens spaces. Kwun [9,10] classified all orientation reversing PL involu- tions of L and all orientation preserving PL involutions of L(p,q), p Egg, which preserve sense and have non-empty fixed point sets. The classification problem of the (sense preserving) free involutions on L(p,q), p > 2), is still open, bUt the prdblem on P3 will be solved by using Rice's work [15]. It will be shown that, up to PL equivalences. there are exactly three PL involutions on P3. Let Mi (i = 1,2) be oriented 3-manifolds and hi be involutions on “1' If there is a suitable invariant 3-cell l in each Mi' by taking the connected sum M1 # M2, along the 3-cells, one can define an involution, denoted by hl # h2, on M1 #M2 induced by hl and h2' The connected sum M1 # M2 is obtained by removing the interior of a nice invariant 3-ce11 from each, and then matching the resulting boundaries using an orientation reversing equivariant homeo- morphism. Notice that hl # h2 depends on the choice of the invariant 3-cells along whose boundaries the connected sum is constructed. All orientation reversing involutions on L(p,q) # L(p,§) are known ([7], [13], [16]). We will also investigate some orientation preserving PL involutions on L(p,q) # L(§,§). As a consequence, up to PL equivalences, there are exactly seven PL involutions on P3 # P3. CHAPTER I FIXED POINT SETS In this chapter we shall study the fixed point set of an orientation preserving involution on a lens space L = L(p,q) which preserves sense and has non—empty fixed point set. Lemma 1.1: Let X be a m-manifold which has a con- tractible universal covering space. If H1(X) is of rank 2 n and there is a short exact sequence O-oA-fi H1(X) S Zp 4 0 where A is a free abelian group of rank n, then H1(X) is a free abelian group of rank n. Proof: Let be a basis for A. Since [ai]i = l,2,---,n f is a monomorphism, we simply identify A 'with the image f(A). Let t be an element of' n1(X) such that g(t) gen- erates Zp' Then H1(X) is generated by the ai and t. Denote the image of an element e of nl(x) by 5 under the natural homomorphism of H1_ to H1. Let Q be the ration- als. H1(X7Q) = H1(X) 3 Q is generated by the 51 ® 1 and t ® 1. Since g(tp) = 0, tp E A, and tp = pt is gener- ated by the Si (notice that we shift from the multiplica- tion notation '25 to the additive notation pt as H1(X) is abelian) . Hence pt® 1 E ([51 ® 1}) which is Q-submod- ule of H1(X) ® Q generated by the Si ® 1. Hence, 'E e 1 e ((51% 11>, and ((51 e 1D = H1(X) a Q. Since 3 H1(X) ® Q is a vector space over Q of rank 2 n, {Si ® 1] is a basis for H1(X) 8 Q. Since A is a normal subgroup k. _ n of H (X), t la.t = H a.J for some k.s'. Abelianizing l n 1 j=1 j n 3 it, a. = k.a.. Hence -. l = k. 5. ® 1 . Hence 1 3&1 J 3 a1 ‘8 32:31 3( J ) ki = 1 and kj 0 if j # i. Therefore, n1(X) is abel- ian. Since 0 4 A 4 n1(X) 4 Zp 4 O is exact and Q is torsion free, A ® Q = H1(X) 8 Q. Since H1(X) is a finite- ly generated abelian group, nl(X) is of rank n. But no non-trivial finite group can act freely on a finite dimen- sional contractible space (due to P.A. Smith [4], 287). Therefore, n1(X) has no torsion subgroup. This completes the lemma. Definition 1.2: Let M1 and M2 be PL manifolds. Two PL homeomorphisms hi on Mi (i = 1,2) are called PL equivalent if there is a PL homeomorphism t of M onto M such that h t = th In this case t is 1 2 2 1' called PL equivariant with respect to h1 and h2° We some- times denote the fact by h1 ~ h2. When hi (i = 1,2) happen to be involutions on M. obviously any equivariant map t sends the fixed point 1' set of h onto the fixed point set of h l 2' Definition 1.3: Let h be an involution on a space M. The quotient space M/Z2 of M generated by h is called the orbit space of h and the projection g: M 4 M/Z2 is called the orbit map of h. ‘We denote the fixed point set of h 'by Fix (h). The following theorem is due to Stallings [17]. Theorem (Stallings): If M is a compact irreducible connected 3-manifold, and if‘ n1(M) has a finitely gener- ated normal subgroup K different from Z whose quotient 2! group is Z, then M is the total space of a fiber space with base space a circle and with fiber a connected 2-mani- fold T embedded in M. whose fundamental group is K. Let D2 be the unit disk in the Gaussian plane of com- plex numbers and 81 its boundary. D2 x S1 is a solid torus whose points can be denoted by (pzl,zz) where 1 21,2 6 S and O s p s 1. 2 Lemma 1.4: The orbit space of a free PL involution h on D2 x S1 is homeomorphic to a disk bundle over $1, and h is PL equivalent to an involution h1 given by either h1(pzl,zz) = (pzl,-zz) or h1(pzl,zz) = ( p21,-22). Proof: Since h is free, the orbit space D2 x 81/22 is a connected orientable compact 3-manifold with boundary. 2 1 _ _ x S /Z2 93 — O and p0 — 1. Since D2 X 51/22 is covered by D2 x 81, we have a short Hence the Betti numbers of D exact sequence 0 4 Z 4111(D2 x Sl/ZZ) 4 Z2 4 0. Since X1) x 81/22 where g and g’ are the orbit maps of h and h’ res- pectively. Since g#[I11(D2 x 81)] = ZZ c: 1'11(D2 x 51/22) = Z, by the lifting theorem, we have a PL homeomorphism t of 2 D x S1 which makes the above diagram commutes. It follows that th = h’t. This completes the Lemma. Remark 1.5: Let h be a PL involution of a finite tri- angulated n-manifold M1. It can be shown that h becomes simplicial after a suitable subdivision such that the fixed point set of h is a subcomplex of the subdivision M2. Let M be the second barycentric subdivision of M2. Then it is easy to check the following properties: (1) F is full sub- complex of M (2) the orbit map 9 of h and the orbit space of h are simplicial and g maps each simplex homeo— morphically. The following result seems to be well-known and freely used by various authors ([10], [19]). For the sake of com- pleteness, we give a proof. Lemma 1.6: Up to PL equivalences, there exists exactly one PL involution h of D2 x S1 with the center circle as the fixed point set. Proof: We first show that the orbit space of h is a solid torus. Let M 'be a triangulation of D2 x S1 as in Remark 1.5 and U be the simplicial neighborhood of the center circle F in M. Then U m D2 x S1 is an invariant neighborhood and U’ = g(U) is a simplicial neighborhood of F’ = g(F) where g is the orbit map of h. Since h is orientation preserving, the orbit space M’ of h is an orientable manifold. Since U’ is orientable, (U’,F’) e (D2 x SI, 0 x 81). 'We want jfl1(M’ - U”) = Z @ Z. Since M- ’~ S1 x S1 x I, X(M-U) = O, and x(M’-U’) = 0. But i Z H2(M’-U’: Q) a H2(M-U: Q) 2 which is 22-invariant homol- ogy with rational coefficient Q (for the proof, see Floyd [2]). By the definition of Z -invariant homology, 2 Z H2(M-U: Q) 2 = {album = a. a e Hzm-U: on 2: {a1h.(a) = a. a e H2(S1 x 51: Q)] where S1 x S1 is the boundary of hd-IJ and h’ = h|S1 x 81. Since h’ preserves the orientation, the induced isomorphism h; is the identity, and H2(M’ - U’; Q) = Q. Hence the Betti number of H2(M’ - U’) is p2 = 1. Since X(M’ - U’) = O, H1(M’-U’) 'is of rank 2. Since M’-U’ is covered by M- U, by Lemma 1.1, n1(M’-U’) = Z @ Z. Therefore, by Stallingsf theorem, L’-U’ is fibered over a circle with fiber T and ‘n1(T)==Z. Since T is a connected 2-manifold, T would be S1 x I or mobius band. But since L’-U’ is orientable, T is orient- able, and T must be S1 x I. Thus L’-U’ may be obtained from S1 x I x I by identifying each (x,0)(x e S1 x I) ‘with (f(x),l) ‘where f is a homeomorphism of S1 x I. Since the number of components of M’-U’ is two, f car- ries Slix i onto S1 x i (i = 0,1). Hence, since M’-U’ is orientable, f must preserve the orientation, and it can 22 be shown that f is isotopic to the identity. Hence M’-U’ S1 x S1 x I. Since U’ is a solid torus, M’ must be a sol- id torus. Let ‘h be the PL involution of D2 x S1 given by 'h(pzl,zz) = (-pzl,zz). Let ‘H be a triangulation of szs1 with respect to_ h’as in Remark 1.5, ‘6 the simplicial neigh- borhood of the center circle ‘3 in ‘M, ‘M’ the orbit space of h and ‘5 the orbit map of B. By the above argument, M’ = D2 x s; U S1 x S1 x I = HY where g(U) = D2 x S1 = 3(5) and S1 x S1 is the boundary of D2 x 81. Since U and V are invariant simplicial neighborhoods of the fixed point sets, one can find invariant 2-cells C and D regularly embedded in U and V as subpolyhedra, respectively. Con- sider the following diagram. ’0 ham-F = (Dz-O) x s1 [g (DZ-O) x81 U SlelxI» ->(D2-0)xsluslxslxr where we will define a PL homeomorphism 1 later. Let (1,0) be generators of' H1(M"F) and n1(fi-P) represented by the path ac and 5D in M and ‘M, respectively. For the sake of briefness, again (1,0) and (0,1) be the canoni- cal generators of’ I11(S1 x 81) ‘which are the fundamental groups of M’-F’ and M’-P’ where P” = g(P) and Sle1 is the boundary of U’ = D2 x S1 = 3’. Without loss of gen- erality, we may assume that (1,0) are generated by g(aC) and 9(aD) of g(aU) and ‘§(3U), respectively. The in- duced homomorphism g# sends (1,0) to (2,0), so that g# (Z a Z) = 22 + ((d,e)) where g#[(0,l)] = (d,e) for some integers d and e, and ((d,e)) is the subgroup of 111(81 x 81) generated by (d,e). Since 9 is the double covering projection, ((d,e)) can not be contained in ((1,0)). Moreover, ((1,0)) n ((d,e)) = [0] since other- wise <(d,e)> c ((1,0)). Hence <(1,0)> + ((d,e)) = <(1,0>@ <(d.e)>. and g#(z e 2) = 22 e <(d.e)>. Since [n1(slxsl): 22 e <(d.e>>J = 2. [I11(Slx81)= z e <(d.e)>J = 1. and n1(S1 x 81) = Z ®‘<(d,e)>. Since g(C) and g(D) are reg- ularly embedded in U’ and H’ as subpolyhedra, there ex- ists a PL homeomorphism q’ of U’ onto V’ carrying g(C) onto ‘§(D). Hence there exists an extended PL homeomorphism q of ‘M’ onto H’. Therefore, q# sends (1,0) to (1,0) 10 and (d.e) to (a.b>. and q#) = z o <(a.b)> = n1(S1 x 81). We may assume b = 1. Define a PL homeomor— . ~ 2 1 ' -a phism t’ of V’ = D x S by t’(pzl.zz) = (pzl.z2 ,22). Since M’ = D2 x S1 U S1 x S1 x I, there exists an exten- sion t on ‘H’. Now define L = tq. Let m be the nice 1 path generating (0,1) of’ I11(S1 x S ) ==n1(fi’ - P’). Con- sidering the action of ‘h and the fact that V is the in- variant simplicial neighborhood of P, g_ (m) is a disjoint union of two simple closed curves. Denote one of them by m’. Let a be an element of’ n1(HF-F) represented by the path cp’. Then §‘#[(1,0)] = (2,0) and §#(a) = (0,1), and ((1,0)) n = [0]. Suppose the contrary that <(l,0)> @ (d) is a proper subgroup of ‘nl(M-V§). Then since 34 is monomorphism, ((2,0)) @‘<(0,1)> is a proper subgroup of g#[n1(FI-'F")J. But [n1~<§m-F))=§#arlm-F>)J= 2. There- fore [’§’# (ulnar-r): <(2,0)> e ((0,1)>] = 1. This is a con- tradiction. Let us look at the following diagram which is a concentration of the work done so far. n1(M-F)=zez snlm-r‘)=ze (1,0) (001) (1,0) (I .# J i .# V1 1(2.0) z, .11 1 (2.0) (0.1) H1(S xS)=Z®Z $H1(S xs=zez 1 ( 0 and L#g#(0,l) = (0,1). Hence by the lifting theorem, we Since t# = -i ), one can check that L#g#[(l,0)]==(2,0) have a PL homeomorphism f which makes the following diagram commute 11 l f 1119(1)2 - 0) x s l i... g ,9 i’ 1 1 'Y 1 (D2- 0) x s )(02 — 0) x s ~--—->(D2 - O) x Slg’h’ 1 = gm)2 - 0) x sl). EHDZ - 0) x 51). ‘where (D2 - O) x S Hence fh ='hf. One can extend f to D2 x S1 in an obv- ious way such that fh = hf on D2 x 51. This completes the lemma. Let h be an orientation preserving PL involution on a lens space L = L(p,q) ‘which preserves sense and has non- empty fixed point set F. By the dimensional parity theo- rem, each component F of F is of l-dimension, and F O is a simple closed curve. Let U be a regular neighbor- O hood of F0 such that U n F = F0. ering projection g: S3 4 L. By the lifting theorem, we bConsider the usual cov- have a PL involution 'h:(S3,yO) 4 (S3,y0) 'where g(yo)(EFd Suppose h is sense preserving. Then g-1(F ) is connect- -1 O ed, and ‘F = 9 (F0) is the fixed point set of 3} By Waldhausen [20], F is an unknotted simple closed curve. Hence 33 - g-1(U) is a solid torus, and. L-U is a solid torus. An explicit argument of the above may be found in [10]. Theorem 1.7: If h is an orientation preserving PL involution of L = L(p,q), p even, which preserves sense and has non-empty fixed point set F, then F is a disjoint union of two simple closed curves. 12 Proof: By the above discussion, L = D2xS1 Uk Sl)(D2 such that D2 x S1 is an invariant regular neighborhood of a component of F for an attaching map k of S1 x 81. Denote hID2 x S1 and hls1 x D2 by hi and hé, respect- ively. Suppose the contrary that Fix (hé) = ¢. Define h1 on D2 x S1 by h1(pzl,zz) = (-pzl.zz). By Lemma 1.6, there exists a PL homeomorphism t’ on D2 x S1 such that h1 = t’hit’-1. Define t of D2 x S1 Uk S1 x D2 onto D2 x S1 Ukt’-1 S1 x D2 by the following: t = t’ on D2 x S1 l 1 2 t = identity on S x D It is a well defined PL homeomorphism. Define h’ on 2 1 l 2 D xs u -15 xD by K=tht"1. Since h1=t’h]’_t"l, kt’ h|D2 x51 = h1 and ‘h|Sl x D2 = hé. It is checked that h is a PL involution, and h and h' are PL equivalent. Hence 'we may assume that h(pzl,22) = (-pzl.zz) on D2 x 51. By Lemma 1.4 and the similar argument as the above, we may fur- ther assume that h(zl,pzz) = (-zl,pzz) on S1 x D2 since Fix (h) = ¢. That is, we may assume that L=D2 x 51 Uf 81 x D2 for an appropriate attaching map f of S1 x S1 and h is given by h(pzl,zz) = (-pzl,zz) on D2 x S1 and h(zl,pzz) (-zl,pzz) on S1 x D2. Consider the following commuta- tive diagram f 81 X 51 A/ 51 X Sl Cpl [ cp f / X A2 D2 x S1 81 x D2 V L(p.q) where m1 and $2 are the inclusion maps. Let (1,0) and (0,1) be the canonical generators of 1'11(S1 x 51) such that f (1,0) = (a,b) and f (0,1) = (c,dL # # where f# is the isomorphism induced by f (we disregard the base point as I11(S1 x Sl,*) is abelian). We may as— sume that I: 3| = 1. One can show that by Van Kampen theorem. H1(L) = [mBl Bc= a. Ba = 1} = (B! Ba= 1} where a and B are the canonical generators of’ I11(D2 x 51) and 111(8l x D2), respectively. Since n1(L(p,q)) = Zp, a = ip, A and a is even. Let g and g be the orbit maps of MD2 x S1 and h]S1 x D2, respectively. Then by Lemma 1.4 and the proof of Lemma 1.6, g(D2 x 81) and 3(51 x D2) are solid tori. Consider the following diagram f D2 x s1 a s1 sl---——————,ls1 x s1 c s1 x 132 I I A g X l/ f ’ D2x813S1x51 Aslxslcslxn2 LQ . A . where g’ and 8’ are induced by g and g, respective— ly and f’ is the induced attaching map in the orbit space A of h. Notice that g#(r,s) = (2r,s) and g#(r,s) = (2r,s) l 1 for any element (r,s) E H1(S x S ). Let fé[(l,0)]= urfibfi 14 and f#[(0,l)] = (c’,d’). By chasing the above commutative diagram, easy computation shows that b = 2b’, and b is even. Since a is even, we have a contradiction to the fact ad - bc = 1. Therefore, Fix(hlS1 x D2) can not be empty. By Tollefson [l9], Fix(hlSl x D2) is a simple closed curve. This completes the proof. CHAPTER II PL INVOLUTIONS ON SOME 3-MANIFOLDS In this chapter, we will investigate all orientation preserving PL involutions on L(p,q), p even, which pre- serve sense and have non-empty fixed point sets. Kwun [9] considered all orientation reversing PL invol- utions on lens spaces, and proved that no lens space except the projective 3-space P3 admits an orientation reversing PL involution and there exists exactly one orientation re- versing PL involution on P3, up to PL equivalences. In this case, the fixed point set is a projective plane P2 plus an isolated point. Using this result applied to Kim and Tollefson's work [7] and Showers"work [16], the orientation reversing PL involutions on the connected sum of two lens spaces are easily classified. Myung [13] initiated this problem and gave a partial solution. We will also study all 3 as well as orientation preserving PL involutions on P3 # P those on P3. Kwun [10] also showed that up to PL equival- ences, there is exactly one orientation preserving PL involu- tion h on L = L(p,q), p ggg, which preserves sense and has non-empty fixed point set if L is symmetric and there are exactly two such h if L is non-symmetric. In the latter case, the two different orbit spaces are L(p,q’) and L(p,q”) where 2q’ 2 i q and 2q”q a *1 mod p. In either case, Fix(h) is a simple closed curve. 15 16 Let (1,0) and (0,1) be the canonical generators of fll(Sl x 81) and k be a PL homeomorphism on S1 x S1 such that k#[(1,0)] = (a,b) and k#[(0,l)] = (c,d). We may assume I: g] = 1 and a 2 O. l 2 2xSJ'U-kS xD Definition 2.1: Define Lk(a,c,b,d) = D where l: E] = 1 and a 2 0. We sometimes denote Lk(a,c,b,d) by Lk if no confusion arises. By Mangler [11], the isotopy classes of homeomorphisms of S1 x S1 are precisely the automorphism classes of 111(81 x 81). Hence, the integers a,b,c and d completely determine the isotopy class of k in Definition 2.1, and hence the homeomorphic type of Lk(a,c,b,d). As Kwun [10] pointed out, if a = 0, Lk is homeomorphic to S1 x 32, if a = l, Lk is homeomorphic to $3, and if a > 1, Lk is homeomorphic to L(a,b). Recall that L(p,q) is homeomorphic I to L(p,q’) ifandonly if q a i q or qq’ a 1:1 modp [12,14]. Lemma 2.2: Let h be a PL involution of Lk(aoC.b.d) such that h(D2 x 51) = D2 x S1 and h is given by 2 h(pzl,zz) = (“921.22) on D x S1 and h(zl,p22)==(zl,-pzz) on S1 x D2. Then the orbit space of h is homeomorphic to Lk,‘(%,c,b,2d), and a is even, where k’ is the attaching map induced by k. Proof: By Lemma 1.6, the orbit space of hID2 x S1 2 and hISl x D are solid tori. Hence the orbit space of 17 h is homeomorphic to Lk,(p’,c’,b’,d’) for suitable k’, p’,b’,c’ and d’. Consider the following diagram. k szslzslxsl >slxslcslxn2 9i 9’ , . D2 xSlZDSlel k >Sl'..x SlcS1 xD2 where g and g’ are induced by the orbit maps of h]D2xS1 and h|S1 x D2, respectively. Notice that g#[(l,0)] = (2.0). g#[(0.1)] = (0.1). g;[(1.0)] = (1.0) and g#[(0.1)]= (0,2). Easy computation shows that p = 2p’, b = b’, c==c’ and 2d = d’. Hence the orbit space of h is homeomorphic to Lk,(§,c,b,2d). This completes the lemma. Definition 2.3: Let p be even and a homeomorphism f of S1 x S1 be given by f(zl,22) = (z§z§,z?zg). Define an involution of Lf(p,c,b,d) by h(pz1,zz) = (-pzl,zz) on 2 l l 2 D x S and h(zl,pzz) = (21,-pzz) on S x D . ‘We denote the involution by h(p,c,b,d). In the above definition, since p is even and b is odd (recall that pdebc = 1), one can easily check that h = h(p,c,b,d) is compatible with the attaching map, i.e., fh = hf. Lemma 2.4: Let hi(i = 1,2) be PL involutions on L. = Lf (a,c,b,d) such that hi(D2 x 81) = D2 x 51, and i J. 1 hi(le.zz) = (-le.22) on 92 x s and hi(zl,pzz) = 1 2 (21,-p22) on S x D . Then hl and h2 are PL equivalent. 18 Prgof: By Lemma 2.2, the orbit spaces of hi are homeomorphic to Lf,(%,c,b,2d) for some attaching maps f1. i Hence f1 and f’ are isotopic, and there exists a level 2 preserving PL homeomorphism H of S1 x S1 x I such that I _ I _ 0 o u 1 l o H1 1 — f2 and H0 — identity. Since S x S x I 18 a boundary collar of S1 x D2, there exists a PL homeomorphism t’ of S1 x D2 c L , onto S1 x D2 c L , such that t’f’== f1 f2 1 I ' ' . _ f2. Define a homeomorphism t1. Lfl 4 Lfé by t1(pzl,zz)-— 2 1 (pzl.zz) on D x S and tl(zl,pzz) = t’(zl,pzz) on S1 x D2. One can check that t1 is well defined. Consider the following diagram. . 2 . 1‘149 L1 'F 11 F12”"""“‘“““_ I'2 — {F21 F 2 2G h2 I igl 392 ' t1 I _ I _ I ..._...,.._-,...-._ _ I _ I Lfi F11 F12 '9 sz’ F21 F22 where Fij (j = 1,2) are the components of the fixed point set Fi of hi' gi are the orbit maps of hi and _ I Since S1 x D2 is a regular neighborhood of Pi , 2 _ _ 2 l _ _ 2 _ 1 Fi)—II1(D xs Fil)—II1((D 0) x5). 1 2 Since t1 is identity on D2 x 81, by the lifting theorem, II‘Li'Fi we have a lifting t such that the above diagram computes. One can extend t to whole L1 in an obvious way such that thl = h2t on L1. This completes the proof. 19 Remark 2.5: Let h be an orientation preserving PL in- volution h of L = L (p,q) , p even, which preserves sense and has non-empty fixed point set.’ By Tollefson [19], any orienta- tion preserving PL involution on S1 x D2 ‘with non—empty fixed point set is PL equivalent to the involution h’ giv- en by h?zl,pzz) = (21,-pz2). Therefore, by using same technique as in the proof of Theorem 1.7, we may assume that L = D2 x S1 Uf S1 x D2 for an appropriate attaching map f and h is given by h(pzl,22) = (-pz1,zz) on D2 x s; and h(zl,pzz) = (21,-pzz) on S1 x D2. Hence, by Lemma 2.4, we may assume h = h(p,c,b,d) on Lf(p,c,b,d) where f is . _ p c b d . given by f(zl,22) — (2122,2122). Since Lf(p,c,b,d) ~ L(p,b), b E iq or bq a 11 mod p. By Lemma 2.2, the or- bit space of h is homeomorphic to L(gub) where b E iq or bq - i1 mod p. Proposition 2.6: h = h(p,c,b,d) can be extended to an effective circle action. Proof: For each Z 6 81, define Sl-action by _ 2 l _ z.(pzl,z2) _ (pzlz,zz) on D x S and z ~(zl.pzz) - (zlzP,pzzzb) on S1 x D2. Remark 2.7: If an involution h of L(p,q) can be extended to an effective circle action, h must be clearly sense preserving. By Proposition 2.6, h(p,c,b,d) is sense preserving. Therefore, by Remark 2.5, the classification prOblem of orientation preserving PL involutions of L(p,qL 20 p even, which preserve sense and have non-empty fixed point sets is the same problem as the classification of those h(p,c,b,d) for various possible c,b,d with pd-cb = 1. The information that we have is that bq a i]. or b ic; mod p. Now we analyze the involution h(p,c,b,d). If h(p,c,b,d) is equivalent to h(p,c’,b’,d’), we denote the fact by h(p.C.b.d) ~ h(P.C’.b’.d’)- Lemma 2.8: For any integers c,b,d with pd-cb = l, (l) h(p,b,c,d) ~ h(p,c’,b,d’) for any integers c’ and d’ with pd’-c’b = l. (2) h(p,c,b,d).~ h(p,c,b+mp,d+mc) (3) h(P.C.b.d) ~ h(P.-C.-b.d) (4) h(P,C,b,d) ~ h(pI-bo-cod) Proof: We will define a homeomorphism t: Lf 4 Lf, where Lf = Lf(p,c,b,d) and LE, to the equivalent involution claimed in (i), i = 1,2,3,4. is the space corresponding In (1), since pd-bc = l : pd’ - bc’, c’ = c+mp, d’= d+mb for some integer m. Define t: Lf 4 Lf, by t(pz1.22) = -m 2 l (pzlz2 ’22) on D x S and t(zl,p22) — (zl,p22) on 1 2 . S x D . For (2), difine t. Lf 4 Lf, by t(pzl,zz) _ 2 _ m (pzl,zz) on D x S and t(zl.p22) - (zl,pzzzl) on 1 2 . s x D . For (3), define t. Lf 4 Lf, by t(pzl.22) — -l 2 l _ -1 l 2 (pzl.z2 ) on D x S and t(zl,pzz) — (zl,pz2 ) on S )cD. For (4), define t: Lf 4 Lf, by t(pzl.22) = (zz,pzl) on D2 x S1 and t(zl.pz2) = (pzz.zl) on S1 x D2 such that 21 t(D2 x 81) = S1 x D2. It is checked that those t are well defined and equivariant homeomorphisms. This completes the proof. Now we are in a position to state our main theorem. Theorem 2.9: Up to PL equivalences, there is exactly one orientation preserving PL involution on L(p,q), P even, which preserves sense and has non-empty fixed point set. Proof: By Remark 2.7, we will consider two involutions h = h(p,c,b,d) and h = h(p,c’,b’,d’). Let L = 1 2 l Lf(p,c,b,d) and L2 = Lf,(p,c’,b’,d’) corresponding to h1 and h respectively. Since L z L(p,b) and L m 2' 1 2 L(p,b’), b 5 ib’ or bb’ :— ilmod p. If b s ib’mod p, b = ila’+ mp for some integer m. By (1), (2), and (3) h1‘~ h2. Suppose bb’ a $21 mod p. Since pd - bc = 1, b’ = i<2+ mp for some m. By (1), (2), (3), and (4), again h1 ~ h2. This completes the theorem. Now consider free Zz-action h on p3. The orbit space M of h is a closed 3-manifold. Since we have a universal covering projection S3 4 P3 4 M, the order of film) is 4. and IIl(M) =z @922 on 2 . Epstein [1] 2 4 completely determined all possible abelian groups which can be fundamental groups of closed 3-manifolds: Z, Z @ Z a z, Z a 22, and Zr' Hence H1(M) should be Z4. Hence 83/24 = M. Rice [15] discussed free Z4-action on 83. As a consequence of the discussion, M = L(4,1). Since every 22 involution on P3 is sense preserving, by Theorem 2.9 and the above discussion, we have the following Corollary. Corollary 2.10: Up to PL equivalences, there is exact- ly one orientation preserving PL involution h on P3 with Fix(h) # ¢ and there is exactly one free involution on P3. Definition 2.11: L(p,q) is called symmetric if i1 mod p. .Q m Since L(p,q) and L(p,q’) are homeomorphic if and I only if q’ a TC} or qq s 11 mod p, L being symmetric is a topological property. Let h be an orientation preserving PL involution on a lens space L(p,q), p even, which preserves sense and non-empty fixed point set. By Theorem 1.7, Fix(h) is a dis- and F . We 1 2 ask that under what condition there exists a PL equivalent joint union of two simple closed curves F homeomorphism t with respect to h such that t(Fl)==F2. Indeed, there exists such a t if L(p,q) is symmetric. Furthermore, the converse is true; By Theorem 2.9, we may assume h = h(p,c,b,d) on Lf(p,c,b,d). If L(p,q) is symmetric, b2 E :hl mod p. Since pd-cb = 1, c = ib+mp for some integer m. By Lemma 2.8, we have the following equivariant mapstti. h(p,c,b,d) = :(p, ib+mp, b,d).E: h(p, b,ib+mp,d) ~2 h(p, b,ib,d-mb)a~3 h(P.ib,b,d-mb) ~4 h(P.ib+mp,b,d) = h(p.c,b,d). Recall that t1(D2 x 31) = l 2 1 2 2 1 S x D and ti(D x S ) = D x S . Let t — t4t3t2tl. 23 Then t is a PL equivartut homeomorphism on Lf(p,c,b,d) such that t(0 x 51) = S1 x 0. Conversely, suppose that there exists a PL equivariant homeomorphism t on L(p,q) such that t(F1) = F2. By the proof of Theorem 1.7, n1(L) = {a,BIE#:= a, Bp = 1} ‘where a and B are represented by the loops F1 and F2, respectively. Let t# be the automorphism induced by t. Then t#(a) = 68 and t#(B)==a5 2 ‘where e = i]. and 5 = $1.. Since t#(Bc) = aéc = 65C 6 3 6c2 2 . and t#(a) = B . B = B , and c a $21 mod p. Since pd - bc = 1, b2 a 11 mod p, which implies L(p,q) is symmetric. Thus, we have the following theorem. Theorem 2.11: Let h be an orientation preserving PL involution on a lens space L = L(p,q), p even, which pre— serves sense and has non-empty fixed point set. Then there exists a PL equivariant homeomorphism t such that t interchanges the two components of Fix(h) if and only if L is symmetric. Let Mi (i = 1,2) be oriented, connected, closed, irreducible 3-manifolds. It is known [7] that a PL involu- tion h on M1 # M2 is either the obvious involution which interchanges M1 and M or of the form h1 # h2 where 2 each hi is a PL involution on Mi' In the latter case, Fix(h) is not empty, and if dim Fix(h) = l, the 2-sphere along which the M1 and M2 are joined meets F in gen- eral position. ‘When each Mi happens to be a lens space and h is of the form hl # h2, it will be convenient to 24 call decomposed sense preserving if h induces the identity of H1(M) = H1(M1) @ H1(M2). ObVlously, if h is decom- posed sense preserving, each hi is sense preserving. In this case, if h is orientation preserving involution with Fix(h) # ¢ and M is symmetric, by Theorem 2.11, h does 1 not depend on how an invariant 3-cell of M1 is chosen to construct hl # h2. Therefore, the following corollary is obtained by using Kwun's result [10] and Theorem 2.9. Corollaryygélg: Up to PL equivalences, there exists exactly one decomposed sense preserving PL involution h on L(p,q) # L(p,q), which preserves the orientation if L(p,q) and L(p,q) are symmetric (p,§ are any integers). There exist exactly two such h if L(p,q) is symmetric and L(p,q) is non-symmetric lens space with 5 odd (p is any integer). Since any involution h on P3 # P3 of the form h1 # h2 is decomposed sense preserving, we have the follow- ing corollary. Corollagy 2.13: Let h be an orientation preserving 3 3 PL involution on P # P . If Fix(h) = ¢ or Fix(h) is connected, h is the obvious involution which interchanges the two P3. If Fix(h) is not connected, Fix(h) is a disjoint union of three simple closed curves and there is exactly one such h, up to PL equivalences. BIBLIOGRAPHY [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] B IBLIOGRAPHY D. B. A. Epstein, "Finite presentations of groups and 3-manifold", Quart. J. Math. Oxford Ser. (2) 12 (1961), 205-212. E. E. Floyd, "Periodic maps via Smith Theory", Seminar on Transformation group by Armand Borel, Ann. of Math. Studies 46, Princeton University Press, 1960, 38. R. L. Fremon, "Finite cyclic group actions on Sl)(82", Thesis, Michigan State University, 1969. S. T. Hu,"Homotopy Theory", Academic Press, 1959, 287. G. R. Livesay, "Fixed point free involutions on the 3-sphere", Ann. of Math. 72 (1960), 603-611. , "Involutions with two fixed points on the three-sphere", Annals of Math. 18 (1963), 582- 593. ‘ P. K. Kim and J. L. Tollefson, "PL involutions on 3- manifolds", to appear. K. W. Kwun, "Piecewise linear involutions of Sl>