A note on 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 $H^*(\text{BDiff}_+^{\delta}(S^{2n-1});\mathbb{Q})$.