Mod-two cohomology of BO(n)

The cases $n=0$ and $n=1$. For $n=0$ the Grassmannian is a point and $H^{\ast }={\mathbb{F}}_2$ by [F1], with no positive classes. For $n=1$ the Grassmannian is ${\mathbb{R}\mathbb{P}}^{\mathrm{\infty }}$, so [F6] gives $H^{\ast }={\mathbb{F}}_2[a]={\mathbb{F}}_2[w_1({\gamma }_1)]$ because $w_1({\gamma }_1)=a$; this is the assertion for $n=1$.

The sphere bundle and its total space. For $n\ge 2$ let $\pi :S({\gamma }_n)\to B\mathrm{O}(n)$ be the unit sphere bundle of ${\gamma }_n$, a numerable fiber bundle with fiber $S^{n-1}$, and consider $p:S({\gamma }_n)\to B\mathrm{O}(n-1)={\mathrm{Gr}}_{n-1}({\mathbb{R}}^{\mathrm{\infty }})$, $(W,v)\mapsto W\cap v^{\perp }$, the orthogonal complement of $v$ in the $n$-plane $W$. In a local frame of ${\gamma }_n$ this is the map $U\times S^{n-1}\to {\mathrm{Gr}}_{n-1}({\mathbb{R}}^{\mathrm{\infty }})$ obtained by orthogonally completing the frame; it is a numerable fiber bundle whose fiber over an $(n-1)$-plane $P$ is the unit sphere in $P^{\perp }\cong {\mathbb{R}}^{\mathrm{\infty }}$, that is $S^{\mathrm{\infty }}$. Since $B\mathrm{O}(n-1)$ is a path-connected CW complex by [F1], the spectral sequence [F4] has $E_2^{a,b}=H^a(B\mathrm{O}(n-1);{\mathcal{H}}^b)$ with ${\mathcal{H}}^b=0$ for $b>0$ and ${\mathcal{H}}^0={\mathbb{F}}_2$ constant by [F5], so the sequence is concentrated in the row $b=0$ and the edge map $p^{\ast }:H^k(B\mathrm{O}(n-1);{\mathbb{F}}_2)\to H^k(S({\gamma }_n);{\mathbb{F}}_2)$ is an isomorphism for every $k$.

The pullback of the tautological bundle splits. Let $L\subseteq {\pi }^{\ast }{\gamma }_n$ be the vertical line bundle, whose fiber over $(W,v)$ is the line $\mathbb{R}v$; it is trivialized by the section $(W,v)\mapsto (W,v,v)$. Orthogonal projection with respect to a metric on ${\pi }^{\ast }{\gamma }_n$ splits ${\pi }^{\ast }{\gamma }_n\cong L\oplus p^{\ast }{\gamma }_{n-1}$, because the fiber of $p^{\ast }{\gamma }_{n-1}$ over $(W,v)$ is $W\cap v^{\perp }$. Therefore, by [F7], the total class is multiplicative, $w({\pi }^{\ast }{\gamma }_n)=w(L)w(p^{\ast }{\gamma }_{n-1})=w(p^{\ast }{\gamma }_{n-1}),$ since the trivial line bundle $L$ has total class $1$; comparing components of this identity gives $w_j({\pi }^{\ast }{\gamma }_n)=w_j(p^{\ast }{\gamma }_{n-1})=p^{\ast }{w}_j({\gamma }_{n-1})$ for every $j$.

The map $\eta$ and surjectivity. Define $\eta :H^{\ast }(B\mathrm{O}(n);{\mathbb{F}}_2)\to H^{\ast }(B\mathrm{O}(n-1);{\mathbb{F}}_2)$ as the composite of ${\pi }^{\ast }$ with the inverse of the isomorphism $p^{\ast }$ of step 1.2. Then step 2.1 gives $\eta (w_j({\gamma }_n))=w_j({\gamma }_{n-1})$ for every $j$, with the convention $w_n({\gamma }_{n-1})=0$. Assume now, as induction hypothesis, that $H^{\ast }(B\mathrm{O}(n-1);{\mathbb{F}}_2)={\mathbb{F}}_2[w_1,\dots ,w_{n-1}]$ with $w_j=w_j({\gamma }_{n-1})$. Then the image of $\eta$ contains all polynomial generators of $H^{\ast }(B\mathrm{O}(n-1);{\mathbb{F}}_2)$ and hence is everything: $\eta$ is surjective.

The Gysin sequence breaks into short exact sequences. The mod-two Gysin sequence of [F3] for $\xi ={\gamma }_n$ is $\cdots \to H^{k-n}(B\mathrm{O}(n))\stackrel{⌣e_2}{\to }{H}^k(B\mathrm{O}(n))\stackrel{{\pi }^{\ast }}{\to }{H}^k(S({\gamma }_n))\stackrel{{\mathrm{\partial }}_G}{\to }{H}^{k-n+1}(B\mathrm{O}(n))\to \cdots ,$ and identifying the middle term with $H^k(B\mathrm{O}(n-1))$ through step 1.2 turns ${\pi }^{\ast }$ into $\eta$. Since $\eta$ is surjective by step 3.1, exactness gives short exact sequences $0\to H^i(B\mathrm{O}(n))\stackrel{⌣e_2}{\to }{H}^{i+n}(B\mathrm{O}(n))\stackrel{\eta }{\to }{H}^{i+n}(B\mathrm{O}(n-1))\to 0$ for every $i$. In particular $⌣e_2:H^i\to H^{i+n}$ is injective for every $i$.

Identification of the Euler class with $w_n$. Take $k=n$ and $i=0$ in step 4.1: the image of $⌣e_2:H^0(B\mathrm{O}(n))\to H^n(B\mathrm{O}(n))$ is a one-dimensional ${\mathbb{F}}_2$-space generated by $e_2({\gamma }_n)$, and it equals the kernel of $\eta$ in degree $n$. That kernel contains $w_n({\gamma }_n)$, since $\eta (w_n({\gamma }_n))=w_n({\gamma }_{n-1})=0$ by step 3.1. Moreover $w_n({\gamma }_n)\mathrm{\ne }0$: let $f:{\mathbb{R}\mathbb{P}}^{\mathrm{\infty }}\to B\mathrm{O}(n)$ be a classifying map of the $n$-fold sum ${\gamma }_1\oplus \cdots \oplus {\gamma }_1$ of the universal line, which exists by [F8]; then $f^{\ast }{w}_n({\gamma }_n)=w_n(f^{\ast }{\gamma }_n)=(w_1({\gamma }_1){)}^n=a^n\mathrm{\ne }0$ by the Whitney formula [F7] and [F6]. Hence both $e_2({\gamma }_n)$ and $w_n({\gamma }_n)$ are nonzero elements of the one-dimensional ${\mathbb{F}}_2$-space $\ker \eta \cap H^n$, so $e_2({\gamma }_n)=w_n({\gamma }_n)$.

Polynomial generation and uniqueness. Let $\phi :{\mathbb{F}}_2[w_1,\dots ,w_n]\to H^{\ast }(B\mathrm{O}(n);{\mathbb{F}}_2)$ send $w_j$ to $w_j({\gamma }_n)$; it is a graded ring homomorphism. Surjectivity is proved by induction on the total degree: for $\xi \in H^k(B\mathrm{O}(n))$, the class $\eta (\xi )\in H^k(B\mathrm{O}(n-1))$ is, by the induction hypothesis on $n$, the image of a unique polynomial $f$ in $w_1,\dots ,w_{n-1}$; then $\xi -\phi (f)$ lies in $\ker \eta$, which by step 4.1 is the image of $⌣w_n$ (using step 5.1), say $\xi -\phi (f)=\zeta ⌣w_n$ for a unique $\zeta \in H^{k-n}(B\mathrm{O}(n))$; by induction on $k$ the class $\zeta$ is the image of a unique polynomial $g$ in $w_1,\dots ,w_n$, so $\xi$ is the image of $f+w_{n}g$. Injectivity is proved by the same decomposition: if $\phi (P)=0$, write $P=f+w_{n}g$ with $f\in {\mathbb{F}}_2[w_1,\dots ,w_{n-1}]$ uniquely; applying $\eta$ gives $0=\eta (\phi (P))$ in $H^{\ast }(B\mathrm{O}(n-1))$, and by step 3.1 and the induction hypothesis on $n$ this is the image of $f$, so $f=0$; then $w_n⌣\phi (g)=0$ and injectivity of $⌣w_n$ from step 4.1 gives $\phi (g)=0$, whence $g=0$ by induction on the degree of $P$. Hence $\phi$ is an isomorphism.

Boundary cases. The case $n=0$ is step 1.1, the case $n=1$ is also step 1.1 and serves as the base of the induction; for $n=1$ the sphere bundle argument is replaced by the published computation $H^{\ast }({\mathbb{R}\mathbb{P}}^{\mathrm{\infty }})={\mathbb{F}}_2[w_1]$. In degree zero both sides are ${\mathbb{F}}_2$, spanned by the unit, and the class $w_0=1$ is the unit by convention. Higher classes above the rank vanish on both sides: $w_j=0$ for $j>n$ by the rank convention and there are no polynomial generators beyond $w_n$. AC is used through [F3] and [F4] and the metric used to split in step 2.1.