Intersection formula for Stiefel-Whitney homology classes (Q1107882)

Generalizing the notion of manifolds, the mod 2 Euler space is defined for a locally compact polyhedron. Besides manifolds, real analytic spaces are typical examples of mod 2 Euler spaces. Let K be a triangulation of an n-dimensional mod 2 Euler space X. Then the sum of all k-dimensional simplexes in the barycentric subdivision of K is a mod 2 cycle of (X,\(\partial X)\). Its homology class is denoted by \(s_ k(X)\) in \(H_ k(X,\partial X;Z_ 2)\) and is called the k-th Stiefel-Whitney homology class of X. Put \(s_*(X)=s_ 0(X)+s_ 1(X)+...+s_ n(X)\) and \([X]=s_ n(X)\). If X is a \(Z_ 2\)-homology manifold, then \(s_*(X)=[X]\cap w^*(X)\), where \(w^*(X)\) is the Stiefel-Whitney cohomology class of X. An n-dimensional Euler space is called of pure dimension if the union of all n-simplexes is dense in X. Theorem. Let M be a PL-manifold. Let X and Y be mod 2 Euler spaces of pure dimension such that they are PL-subspaces and \(\partial X\cup \partial Y\subset \partial M\), (X-\(\partial X)\cup (Y-\partial Y)\subset M-\partial M\). Let f:\(| X\to M\), g: \(Y\to M\) and h: \(X\cap Y\to M\) be the inclusions. If X is transverse to Y, then \(X\cap Y\) is a mod 2 Euler space and \(f_*s_*(X)\cdot g_*s_*(y)=h_*s_*(X\cap Y)\cap w^*(M)\), where \(a\cdot b\) is the homology intersection defined by \(a\cdot b=[M]\cap (([M]\cap)^{-1}a\cup ([M]\cap)^{-1}b)\).