Let $X$ be a manifold and $E\to X$ a complex vector bundle and let's work in $H^\bullet(X,\mathbb{Z})$. Given the total Chern class of $E$, $c(E)=1+c_1(E)+\cdots+c_n(E)$, we can define the total Segre class of $E$ to be $s(E)=1+s_1(E)+\cdots+s_n(E)$ to be the inverse of the total Chern class.
Equivalently, recalling that $c(E)=\prod_{i=1}^n1+\alpha_i(E)$, with $\alpha_i(E)$ the $i$-th Chern root of $E$, we can define $$s(E)=\prod_{i=1}^n\frac{1}{1+\alpha_i(E)}=\prod_{i=1}^n\sum_{k=0}^\infty (-\alpha_i(E))^k.$$
I am interested in the real analogue, now working in $H^\bullet(X,\mathbb{F}_2)$, that is to say the following:
given $V\to X$ a real vector bundle, the total Stiefel-Whitney class of $V$ is $$w(V)=1+w_1(V)+\cdots+w_n(V)=\prod_{i=1}^n1+\sigma_i(V)$$ (where the $\sigma_i(V)$ are the Stiefel-Whitney roots of $V$) and we can analogously define its inverse as $$p(V)=\prod_{i=1}^n\frac{1}{1+\sigma_i(V)}=\prod_{i=1}^n\sum_{k=0}^\infty\sigma_i(V)^k=1+p_1(V)+\cdots+p_n(V).$$
This all seems quite natural but I have never seen it anywhere, the only sources I could find are [1] Stiefel-Whitney Homology Classes and Riemann-Roch Formula of Matsui and Sato and [2] Axioms for Stiefel-Whitney Homology Classes of Some Singular Spaces of Veljan, but they work in a different, more general context.
To conclude, I'm asking if these classes have been studied and what else is known about them.
My main concern is the integral of $p_n(V)$ with [1] and [2] suggesting that $\int_Xp_n(V)=\chi(V)\operatorname{mod} 2$, but I would appreciate more sources.
Best Answer
Unfortunately $\int_Xp_n(V)\neq\chi(V) \operatorname{mod} 2$ in general. In particular $\int_Xp_n(TX)=0$ and it can be shown as follows.
By the multiplicativity of the Stiefel-Whitney classes, given two bundles $V, W$ such that $V\oplus W=\mathbb{R}^d$, then $p_k(V)=w_k(W)$. It is now easy to show that $p_n(TX)=0$ since any $n$-manifold can be embedded into $\mathbb{R}^{2n}$. Calling $\nu$ the normal bundle we have $p_n(TX)=w_n(\nu)=0$, with the last equality given by Corollary 11.4 in Characteristic Classes of Milnor and Stasheff.
I still don't know about the general case.