Stiefel-Whitney homology classes and Euler subspaces (Q1911167)

Given a locally compact polyhedron \(X\) of dimension \(n\), it is a \(\text{mod } 2\) Euler space if there is a cohomology class on \(X\) which restricts to \(1 \in H^n (X,X - x; \mathbb{Z}_2)\) for all \(x \in X\). The \(k\)-th Stiefel-Whitney class of an Euler space is given by the cycle representing the sum of all the \(k\)-simplices. Then the result shown is that any element of homology less than \(k \leq n -1\) can be represented as the image of the \(k\)-th Stiefel-Whitney class of a \(k\)-dimensional \(\text{mod }2\) Euler class imbedded into \(X\).