Is the top Stiefel-Whitney number of a topological manifold the Euler characteristic mod two?

Question

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, $w = \operatorname{Sq}(\nu)$. The expression $\operatorname{Sq}(\nu)$ 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 $\chi(M)$?

Answer 1

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

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 $\chi(M) = \text{rk } H^k(M;\Bbb Z/2) \pmod 2.$ So the claim is that $\langle \nu_k^2, [M] \rangle = \text{rk } H^k(M;\Bbb Z/2) \pmod 2.$

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 $\text{rk } V = y^2 \pmod 2$.

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 $\begin{pmatrix}1\end{pmatrix}$ and $\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}$, for which the respective characteristic vectors are $(1)$ and $(0,0)$.

Answer 2

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 $\mathbb{F}_{2}$ 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.

Answer 3

Elaborating on Mark's comment, associated to a manifold of type CAT=PL,DIFF,TOP or a menagerie of others we have a $\mathbb{R}^n$ bundle with structure group $CAT$. In all these cases, we can form the fiberwise one point compactification, and then identify these to get the Thom space. The normal proof of the Thom isomorphism goes through with the obvious notion of orientability of these bundles.

Recall that to define Stiefel-Whitney classes for a vector bundle, we only use Steenrod operations and the mod 2 Thom isomorphism. Since every disk bundle is oriented mod 2, we can use the exact same definition. As well, if we have an integrally oriented disk bundle we can use the exact same definition of Euler class, as a pullback of the Thom class by the zero section.

An oriented CAT manifold is easily seen to have an integrally oriented $\mathbb{R}^n$ bundle, so we have an Euler class. By picking a section of the disk bundle with isolated singularities (where it hits the zero section), we may mimic the proof in the DIFF case since the total index of the section of the disk bundle is the Euler characteristic of the manifold. This means the Euler class evaluates on the fundamental class to the Euler characteristic.

Then the same proof as the DIFF case shows that the CAT Stiefel-Whitney class is the mod 2 reduction of the CAT Euler class. And, again, the same proof as in the DIFF case shows that Stiefel-Whitney classes can be defined via your formula with the Wu classes, so this definition agrees with yours.