Rational Pontryagin classes and functor calculus

TL;DR

This paper investigates whether the classical relation between Euler and Pontryagin classes in integral cohomology extends to the rational cohomology of BSTOP(2m), using smoothing theory and functor calculus.

Contribution

It reformulates the problem of Pontryagin classes and Euler class relations in rational cohomology within differential topology and functor calculus frameworks.

Findings

Reformulation of the problem using smoothing theory.

Application of functor calculus to the cohomology question.

Insights into the extension of classical classes in rational cohomology.

Abstract

It is known that in the integral cohomology of BSO(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 BSTOP(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 BSTOP(2m). In this paper we use smoothing theory and tools from homotopy theory to reformulate the hypothesis, and variants, in a differential topology setting and in a functor calculus setting.