Stiefel-Whitney homology classes and homotopy type of Euler spaces
A polyhedron X is called an integral Euler space (resp. mod 2 Euler space) if for each x in X the Euler number of the pair (X,X-x) is \((- 1)^ n\) (resp. 1 (mod 2)) [cf. \textit{S. Halperin} and \textit{D. Toledo}, Ann. Math., II. Ser. 96, 511-525 (1972; Zbl 0255.57007)]. A polyhedron X is called purely n-dimensional if the union of all n-simplexes in a triangulation of X is dense in S. If X is a mod 2 Euler space and K' denotes the barycentric subdivison of a triangulation K of X, the sum of all k-simplexes in K' is a mod 2 cycle and defines an element \(s_ k(X)\) in \(H_ k(X; {\mathbb{Z}}_ 2)\) [op. cit.]. The element \(s_ k(X)\) is called the kth Stiefel-Whitney homology class of X. The authors prove: Let X be a purely n-dimensional mod 2 Euler space and let \(a_ i\) for \(i=1,2,...,n-1\), be elements in \(H_ i(X; {\mathbb{Z}}_ 2)\). Then there exists a purely n-dimensional mod 2 Euler space Y and a homotopy equivalence \(h: X\to Y\) such that \(h_*(a_ i)=s_ i(Y)\) for \(i=1,...,n-1\) and \(h_*s_ n(X)=s_ n(Y).\) By making use of the Bockstein homomorphisms associated with the exact sequence \(0\to {\mathbb{Z}}_ 2\to {\mathbb{Z}}_ 4\to {\mathbb{Z}}_ 2\to 0\), they establish an analogous theorem for a purely n-dimensional integral Euler space. The techniques of the proof are exploited to obtain counterexamples to Halperin's conjecture on normally non-singular maps [cf. \textit{W. Fulton} and \textit{R. MacPherson}, Mem. Am. Math. Soc. 243 (1981; Zbl 0467.55005)].
purely n-dimensional integral Euler space
scientific article; zbMATH DE number 3929956