Stiefel-Whitney homology classes and homotopy type of Euler spaces

Abstract

In this paper, we construct Euler spaces in fixed homotopy types such that the Stiefel-Whitney homology classes are equal to given homology elements.As a byproduct, we obtain counterexamples to Halperin's conjecture (Fulton-Let $X$ be a locally compact n-dimensional polyhedron.For a point $x$ in $X$ , let $\chi(X, X-x)$ denote the Euler number of the pair (X, $X-x$ ).The polyhedron $X$ is called an integral Euler space (resp.mod 2 Euler space) if for each $x$ in $X$ , $\chi(X, X-x)=(-1)^{n}$ (resp.$\chi(X,$ $X-x)\equiv 1$ (mod2)) (Halperin and Toledo [6]).Sullivan [9] has shown that complex analytic spaces (resp.real analytic spaces) are integral Euler spaces (resp.mod2 Euler spaces).Let $K'$ denote the barycentric subdivision of a triangulation $K$ of a polyhedron X.If $X$ is a mod2 Euler space, the sum of all k-simplexes in $K'$ is a mod2 cycle and defines an element $s_{k}(X)$ in $H_{k}(X;Z_{2})$ (cf. [6]).Note that, if $X$ is not compact, we consider the homology of infinite chains.The element $s_{k}(X)$ is called the k-th Stiefel-Whitney homology class of $X$ .If $X$ is connected and compact, $s_{0}(X)$ is the mod 2 reduction of the Euler number $\chi(X)$ , where we identify $H_{0}(X;Z_{2})$ with $Z_{2}$ .If $X$ is a smooth manifold, PL-manifold, or $Z_{2^{-}}$ homology manifold, the class $s_{k}(X)$ is known to be equal to the Poincar\'e dual of the Stiefel-Whitney cohomology class $w^{n-k}(X)$ (Cheeger [3], , Taylor [10], Blanton-McCrory [2], Veljan [11], Matsui [8]).Consequently, for such spaces, the Stiefel-Whitney homology classes $s_{*}(X)$ are homotopy type in- variant. For urther properties of Stiefel-Whitney homology classes, see [1], [7].A polyhedron $X$ is called purely n-dimensional if the union of all n-simplexes in a triangulation of $X$ is dense in $X$ .We have the following concerning mod2 Euler spaces: THEOREM 1.Let $X$ be a purely n-dimensional mod2 Euler space and let $a_{i}$ , for $i=1,2,$ $\cdots$ $n-1$ , be elements in $H_{i}(X;Z_{2})$ .Then there exist a purely n- dimenstonal mod2 Euler space $Y$ and a homotopy equivalence $h:Xarrow Y$ such that $h_{*}(a_{i})=s_{i}(Y)$ for $i=1,2,$ $\cdots$ $n-1$ and $h_{*}s_{n}(X)=s_{n}(Y)$ .