Rational Pontryagin classes and functor calculus

Abstract

It is known that in the integral cohomology of $B\mathrm{SO}(2m)$, the square of the Euler class is the same as the Pontryagin class in degree $4m$. Given that the Pontryagin classes extend rationally to the cohomology of $B$STOP($2m$), it is reasonable to ask whether the same relation between the Euler class and the Pontryagin class in degree $4m$ is still valid in the rational cohomology of $B$STOP($2m$). In this paper we reformulate the hypothesis as a statement in differential topology, and also in a functor calculus setting.