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.



Real analogue of Segre classes
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.


