= Rational odd-rank stabilization of orthogonal classifying spaces
{title2=$H^*(BO(n);\mathbb Q)\cong H^*(BO(n-1);\mathbb Q)\quad(n\text{ odd})$}
The stabilizer of a line in $O(n)$ is $O(1)\times O(n-1)$, giving the <homogeneous space> $\mathbb{RP}^{n-1}$. Its classifying-space <Serre fibration> has rationally acyclic fiber for odd $n$, so its cohomological <Serre spectral sequence> has only its degree-zero row and gives the displayed restriction isomorphism. The factor $BO(1)=\mathbb{RP}^\infty$ is itself rationally acyclic. For even $n$, the fiber has a top rational class, with determinant-sign monodromy, so the one-row argument fails. The top universal <Pontryagin class> $p_{n/2}$ is nonzero on $BO(n)$ but vanishes on restriction to $BO(n-1)$, witnessing genuine failure of stabilization.
Back to article page
