Vtyjkyuk

Recall that the Stiefel-Whitney classes of a smooth manifold are defined to be those of its tangent bundle - this definition doesn't extend to topological manifolds as they don't have a tangent bundle. Wu's theorem states that for a closed smooth manifold, $w = operatornameSq(nu)$. The expression $operatornameSq(nu)$ makes sense for a closed topological manifold and therefore serves as a definition for the Stiefel-Whitney classes on such a manifold.

Recall that if $M$ is a closed smooth $n$-dimensional manifold, then $w_n(M)$ is equal to the mod $2$ reduction of $e(M)$, see Corollary 11.12 of Milnor and Stasheff's Characteristic Classes. In particular, the Stiefel-Whitney number $langle w_n(M), [M]rangle$ is the mod $2$ reduction of the Euler characteristic. Is this still true for closed topological manifolds?

Let $M$ be a closed topological $n$-dimensional manifold. If $w_n(M)$ is the top Stiefel-Whitney class of $M$, as defined above, is the Stiefel-Whitney number $langle w_n(M), [M]rangle$ the mod $2$ reduction of $chi(M)$?

As you say, we define $w_n = sum textSq^i nu_n - i$, where $nu_n-i$ is the Wu class, the class such that $nu_n-i cup c = textSq^n-i c$ for $c in H^i$. So as a corollary we have $textSq^i nu_n - i = nu_i cup nu_n-i$.

Because $nu_j$ vanishes for $j > n/2$, the sum over $i$ is only the term $nu_n/2^2$ when $n$ is even, and $0$ when $n$ is odd. As the Euler characteristic of an odd-dimensional closed manifold vanishes, this gives the odd-dimensional case.

Now when $n = 2k$, using Poincare duality mod 2 we see that $chi(M) = textrk H^k(M;Bbb Z/2) pmod 2.$ So the claim is that $langle nu_k^2, [M] rangle = textrk H^k(M;Bbb Z/2) pmod 2.$

This is because $nu_k$ is a characteristic vector for the symmetric bilinear cup-product form on $H^k(M;Bbb Z/2)$; in fact, for any 'characteristic vector' $y$ for a nondegenerate symmetric bilinear form over a $Bbb Z/2$-vector space $V$, meaning that $y cdot x = x^2$ for all $x$, we have $textrk V = y^2 pmod 2$.

The most obvious way for me to see this is to classify nondegenerate symmetric bilinear forms over $Bbb Z/2$ vector spaces: they are all sums of copies of $beginpmatrix1endpmatrix$ and $beginpmatrix0 & 1 \ 1 & 0endpmatrix$, for which the respective characteristic vectors are $(1)$ and $(0,0)$.

It has been proved in the preprint (page 6) by Renee Hoekzema that the vanishing of the $w_n(M)$ implies $chi(M)$ is even. The proof uses the fact that a symplectic vector space over $mathbbF_2$ has even dimension. It is quite similar to the one Mike Miller given here without the induction procedure.

The author suggests there is a more direct proof generalizing the one from Milnor-Stasheff without using the Euler class. I am not sure it might be. The paper actually proved much more and I found it really interesting.