A note on invariants of foliated 3-sphere bundles

TL;DR

This paper demonstrates that the rational cohomology of the classifying space for SO(4) injects into the group cohomology of orientation-preserving diffeomorphisms of the 3-sphere, revealing new connections between bundle invariants and diffeomorphism groups.

Contribution

It establishes an injection of the rational cohomology of BSO(4) into the group cohomology of Diff^+(S^3), extending understanding of invariants of foliated 3-sphere bundles.

Findings

H^*(BSO(4);Q) injects into H^*(Diff^+(S^3);Q)

Monomials in Euler and Pontrjagin classes are nontrivial in certain group cohomologies

Provides new insights into invariants of foliated 3-sphere bundles

Abstract

In this note we prove that $H^*(\text{BSO}(4);\mathbb{Q})$ injects into the group cohomology of $\text{Diff}^+(S^{3})$ with rational coefficients. The proof is based on an idea of Nariman who proved that the monomials in the Euler and Pontrjagin classes are nontrivial in $H^*(\text{BDiff}_+^{\delta}(S^{2n-1});\mathbb{Q})$.