Inner product spaces associated with Poincaré complexes
In: Transactions of the American Mathematical Society, Jg. 260 (1980), S. 411-419
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We consider the homotopy type classification of a certain kind of Poincare complex. First we define an inner product space associated with such a Poincare complex and we investigate the relation between the inner product space and the homotopy type of the Poincare complex. As an application, some results for manifolds are proved. 0. Introduction. Let M be an oriented closed manifold of dimension 2n. It is well known that the bilinear map Hn(M; Z) x Hn(M; Z) -_ Z, ((x, y) -> x U Y, >) makes Hn(M; Z) an inner product space (in the sense of [2]) if M has no torsion. This inner product space is closely related with the homotopy type of M. For example, J. Milnor has proved in [2, p. 103] the following (cohomology version). THEOREM. Let Mi (i = 1, 2,) be simply connected closed manifolds of dimension 4. Their inner product spaces are isomorphic to each other if and only if they have the same oriented homotopy type. We are interested in a generalization of Milnor's theorem to the case of torsion. In ? 1 we shall define an inner product space over Q/ Z associated with an oriented Poincare complex and in ??2-4 apply this to the homotopy type classification of Poincare complexes K which satisfy the conditions (1) K is (n 1)-connected and of dimension 2n + 1, (2) Hn(K; Z) is a finite abelian group G without 2-torsion (n > 3). We call K such as above a Poincare complex of type Pn(G) and in ?2 we shall discuss a special case G = ESZp,. In ?3 we shall give a decomposition theorem. THEOREM A. Let K be a Poincare complex of type Pn(G) and let G = E Gp' be a direct sum decomposition of G, where Gp is the p '-component of G. Then K has the same oriented homotopy type as the connected sum of Poincare complexes of type P"(GP) (i = 1, 2,..., odd primes p). Finally, in ?4 we prove THEOREM B. Let K and K' be S-reducible Poincare complexes of type Pn(G). Then K has the same oriented homotopy type as K' if and only if their product spaces are isomorphic. Received by the editors December 12, 1978 and, in revised form, April 5, 1979 and October 19, 1979. AMS (MOS) subject classifications (1970). Primary 55DI5. ? 1980 American Mathematical Society 0002-9947/80/0000-0354/$03.25 411 This content downloaded from 157.55.39.136 on Tue, 05 Jul 2016 05:33:57 UTC All use subject to http://about.jstor.org/terms 412 SEIYA SASAO AND HIDEO TAKAHASHI As an application we also prove THEOREM C. Let M be a closed smooth manifold whose underlying Poincare complex is of type Pn(G). Then if n m 3 mod 4 the oriented homotopy type of M is determined by the isomorphism classes of the inner product space associated with M. REMARK. In the case n 3 mod 4, Theorem C also holds for M with trivial Pontriagin class Pn+ l14(M). In particular, for a manifold which is of type Pn(G) we have COROLLARY C-1. If n 0 O mod 2, the oriented homotopy type of M is uniquely determined by Hn(M; Z). COROLLARY C-2. If n =1 mod 4, M is the connected sum # MP,,k up to oriented homotopy equivalence, where Mp,k is a closed manifold whose underlying Poincare complex is of type Pn(Zp,). In ? ? 1-4 we always assume that n > 3 and p is an odd prime number. 1. Inner product spaces. Let K be a simply connected Poincare complex with the fundamental class 11K E H2n+l (K; Z). Let 6: H*(K; Q/Z) -H*+ '(K; Z) be the connecting homomorphism associated with the exact sequence of coefficients, 0-> Z-> Q-> Q/Z->O. We consider the bilinear map A8K: Hn(K; QIZ) x H n(K; QIZ) ->QIZ defined by the composite map Hn(K Q/) Hn(KQ/z) 1xa Hn(K; QIZ) H n+ 1(K; Z) ~~~~~IHn(K; QIZ) x H n(K; QZ) Q/Z, lxD K I> where D is the Poincare duality map and denotes the Kronecker pairing. The following is well known, LEMMA 1.1. 83K(X,Y) = (1)n+ 18K(y, X). Moreover assuming that Hn(K; Z) (and hence Hn +(K; Z)) are torsion, we have PROPOSITION 1.2. 8K is a completely orthogonal pairing (cf. [2]) and symmetric, for odd n, and skew symmetric, for even n. PROOF. Clearly the latter follows from Lemma 1.1. The former follows from the Poincare duality theorem for torsion groups. Thus we have the inner product space V(K) = {Hn(K; Q/Z), 1K) associated with a Poincare complex K. Clearly this inner product space is an oriented homotopy type invariant for oriented Poincare complexes and we are interested in the problem, "Is the converse true?" The next section investigates a special case. 2. A special case G = E Zpi. Let M(n, p') be the Moore space of type (n, Zpi). Note the following easy LEMMA 2.1. Tn+ I(M(n, pi)) = 0 = qJn+2(M(n pi))This content downloaded from 157.55.39.136 on Tue, 05 Jul 2016 05:33:57 UTC All use subject to http://about.jstor.org/terms INNER PRODUCT SPACES 413 To study the homotopy type classification of Poincare complexes which are (n 1)-connected and of dimension 2n + 1 having HJ(K; Z) = E'Zp,, we must investigate the homotopy group 7T2n(Ms) where M, denotes the wedge sum of s copies of M(n, p'). Clearly a Poincare complex K has the same homotopy type as the mapping cone of a map f: S2n -Ms. First we note the following two lemmas which are proved by standard arguments. LEMMA 2.2. The smash product M(n, p') A M(n, p') is the wedge sum M(2n, pi) V M(2n + 1, p') up to homotopy. LEMMA 2.3. T2n(M,) = 7T2n(M_-1) ? T2n(M0) ? 7T2n+1(M A Ml). Secondly we consider the special case s = 2. Let P be the natural map M(n, pi) -> +l = M(n,p,)/Sn and consider the map P = id V P: M(n, p') V M(n, p') -> M(n, p') V Sn+ . Then we have the commutative diagram, 7T2n(M2) = T2n((MI) ? T2n(M1) fl T2n+i(Mi x MI) M2) JP* Xid P* (id x P) 72n(MI V Sn+') = '2n(M1) ? 72n(Sn+l) ED g2n+1(M, x Sn+1, Ml v sn+1), in which (id x P)* is isomorphic because we have isomorphisms 2n+I1(Ml x Ml, M V MI) 4 H2n1(Ml X Ml, M, V MI) l(id xP)* 7t"+(MI x Sn+1' M, V Sn+l) H 2+(m, x sn+1, ml V sn+1) Let t be the inclusion Sn _ M = M(n,p') = Sn U en+1 Since it is easily seen that the Whitehead product [t, /n+,] generates the group 7T2n+I(M1 x S'+' Ml V S"+'), we can take the element a = P -7([t, /n+J]) as a generator for the group a72n+ I(MI x MI, M2) using the above diagram. Let ik be the inclusion of M, into the kth factor of M, and ik,l be the map M2 = MI V MI M, (ik,! = ik V i1). Then by Lemmas 2.2 and 2.3, we have PROPOSITION 2.4. T2n(Ms) = S7r2n((Ml) ? XZPi[ak,J], where the second summation runs over 1 M, onto the kth factor of M, We take {xk = p*(xo)) as a system of generators for H (Ms; Q/Z) and identify Hn(Ms; Q/Z) with Hn(Kf; Q/Z) for the mapping cone Kf of a mapf: S2n -> Ms. Now we define a homomorphism Ds: T2n(M,) -+ {s x s-matrices over Q/Z} by the formula Ds(f) = (fk,l)' fk,l = , where ,p denotes the oriented generator of H2n+I (Kf; Z). This content downloaded from 157.55.39.136 on Tue, 05 Jul 2016 05:33:57 UTC All use subject to http://about.jstor.org/terms 414 SEIYA SASAO AND HIDEO TAKAHASHI LEMMA 2.5. The matrix (fk,l) is symmetric for odd n and skew symmetric for even n. Moreover, Kf is a Poincare complex if and only if D,(fl = (fk,l) is invertible. PROOF. The first part follows from Lemma 1.1. The second part is equivalent to Poincare duality. Let p be a map M, -> M, and define the matrix U = (ukl) (uk! C Q/Z) by p*(Xk) = X1Uk,1X, (k = 1, . . . , t). Then by the definition of D, and U, we have LEMMA 2.6 (NATURALITY OF Dl). For a map p: M, M,, we have D,(9p*(f)) = U o D,((f) o tU (f E Now we prove a key lemma for our purpose. LEMMA 2.7. Let f = Xfk + Xak l[akl] (k Ml (Pk(f) = f,). For fk,l = ak,l, it is sufficient to prove DS(ak l) = {(k, 1) and (1, k) components are + 1 and others are 0), and this is equivalent to D2(a) = (4? ) by the naturality. And so we consider as follows: Now there exists a map X: K. -> Ml x Ml such that XiM2 = idM2: M2 -> M2 C Ml x Ml and X*(uL) is a generator of H2n+1(MI x MI; Z) z Zp,. Then we consider the following diagram
Titel: |
Inner product spaces associated with Poincaré complexes
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Autor/in / Beteiligte Person: | Takahashi, Hideo ; Sasao, Seiya |
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Zeitschrift: | Transactions of the American Mathematical Society, Jg. 260 (1980), S. 411-419 |
Veröffentlichung: | American Mathematical Society (AMS), 1980 |
Medientyp: | unknown |
ISSN: | 1088-6850 (print) ; 0002-9947 (print) |
DOI: | 10.1090/s0002-9947-1980-0574788-5 |
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