# Real analogue of Segre classes

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.

• Noooooo P is for Pontryagin. Dec 7, 2021 at 19:34

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.