A Note on Simple Duality

Certain spaces satisfying Poincare duality are shown to have the homotopy type of simple Poincare complexes. The motivation for and applications of this result are the study of free actions of finite groups on manifolds. Let X be a finite CW complex with finite fundamental group ir. Then X is an n-dimensional Poincare complex if there is a class [X] E H, (X) such that the slant product with the transfer of [X], nixi n] ( -) * ( ) induces a chain homotopy equivalence. Let H be a quotient of Wh(ir) = K,(Z1r)/(?1r) Then X is an H-simple Poincare complex if the torsion im(r(fl[X])) = 0 E H. For example: (i) Wh(ir)-simple Poincare complexes are called simple. Any closed manifold is a simple Poincare complex. (ii) Wh'(ir)-simple Poincare complexes are weakly-simple, where Wh'(ir) = im(Ki (Zir) -K1 (Qir))/(?ir). C. T. C. Wall [7] proved that every odd-dimensional X is weakly-simple. (iii) H. Bass [1] showed that Wh(ir) = Wh'(ir) for ir cyclic. Thus every odddimensional Poincare complex with cyclic fundamental group is simple. 1. Simple duality. Let A = im(Kl(Zir) -Kl(Z(,)7r))/(?7r). Here Z(,) means we invert all primes which do not divide the order of ir. The object of this section is to prove THEOREM 1. 1. Let f: X -Y be a map between n-dimensional Poincare complexes with n even, inducing isomorphisms f*: 7rlX --r 7Y, f*: H* (Xk; Z(,)) )H* (Y; Z(r) ), where ir = 7r1X is finite. Assume that the induced action of ir on H* (X; Z[1/I1rI]) and H*(Y; Z/[1/I1rl]) is trivial. Then X has the homotopy type of an A-simple Poincare complex if and only if Y does. REMARK. In particular the theorem is true if A-simple is replaced by weaklysimple. If the 2-torsion of Oliver's group Cl1 (Zir) vanishes, then K1 (Zir) -A is Received by the editors May 30, 1984 and, in revised form, August 6, 1984. 1980 Mattematic8 Subject Classifiation. Primary 57P10; Secondary 57Q10, 57Q12, 57S17. Key word and phrase8. Simple duality, Reidemeister torsion. 'Partially supported by an NSF grant. (?1985 American Mathematical Society 0002-9939/85 $1.00 + $.25 per page 343 344 J. F. DAVIS AND PETER LOFFLER an isomorphism modulo odd torsion. In this case any A-simple Poincare complex has the homotopy type of a simple Poincare complex. If the 2-Sylow subgroup of ir is cyclic, the 2-torsion of Cll(Zir) vanishes by [5]. Thus we have COROLLARY 1.2. Let f:X -Y as above. If the 2-Sylow subgroup of ir is cyclic, then X has the homotopy type of a simple Poincare' complex if and only if Y does. Our major tool is the Bass localization sequence [1]. Let T be a multiplicative subset of a ring A. By a (A,T)-module we mean a finitely generated T-torsion A-module of homological dimension one. Let K1(A,T) denote the abelian group resulting from the Grothendieck construction on the category of (A, T)-modules. There is an exact sequence K1 (A) K1 (TA) K1 (A, T) I Ko(A) -Ko(T-'A). Let C be a finite dimensional chain complex of finitely generated free A-modules such that Hi(C) is a (A, T)-module for all i. Then the Reidemeister torsion /\ (C) E K1 (T-'A)/K1 (A) = ker(u) is defined by giving C any A-base and computing the torsion of the based acyclic complex T-1A OA C. We now quote from [3]: PROPOSITION 1.3. The Reidemeister torsion A\(C) is equal to the torsion characteristic X(H*(C)) = Z(-1)i[Hi(C)] E keru. The following lemma applies even in the simply-connected case. LEMMA 1.4. Let g: V -W be a map between n-dimensional Poincare' complexes with n even which induces an isomorphism g*: H* (V; Q) -H* (W; Q). If the Euler characteristic x(W) = 0, then 7 IHi (g) = J Hi (g)I. ieven , iodd REMARK. If the degree of g were one, then this would simply follow from Poincare duality since then Hi(g) Hni+(g). For general degrees this does not hold, so different techniques are necessary. PROOF OF 1.4. Applying the previous proposition with A = Z, T = Z 0, it suffices to prove A\(g) = 0 E Ki(Z, Z 0). Equivalently we show f(g) = 0 E K1(Q) = K1(Q)/K1(Z) K1(Z, Z 0). If R is a ring, and a: M -N is a homomorphism of Z-modules we write aR for a 0 idR. Choose fundamental classes [V], [W], and an integer k such that g# [V] = k[W] E Cn (V; Q). Consider the following commutative diagram: Cn* (W; Q) Q C (V;Q)