Fibrewise rational H-spaces

TL;DR

This paper extends classical theorems of Hopf and Leray-Samelson to fibrewise rational H-spaces, showing their triviality after rationalization and applying these results to universal adjoint bundles, impacting finiteness theorems.

Contribution

It introduces fibrewise versions of key classical theorems, enabling new insights into the structure of fibrewise H-spaces and their applications to existing finiteness results.

Findings

Fibrewise versions of Hopf and Leray-Samelson theorems proved.

Fibrewise H-triviality after rationalization established for certain fibrewise H-spaces.

Application to universal adjoint bundles and existing finiteness theorems.

Abstract

We prove fibrewise versions of classical theorems of Hopf and Leray-Samelson. Our results imply the fibrewise H-triviality after rationalization of a certain class of fibrewise H-spaces. They apply, in particular, to universal adjoint bundles. From this, we may retrieve a result of Crabb and Sutherland [Proc. London Math. Soc. (2000), 747-768], which is used there as a crucial step in establishing their main finiteness result.