On invariants of foliated sphere bundles

TL;DR

This paper investigates invariants of flat sphere bundles, proving non-vanishing of certain characteristic classes and answering a question by Haefliger about flat odd-dimensional sphere bundles.

Contribution

It demonstrates that specific characteristic classes remain non-trivial in certain cohomology groups, and shows that some sphere bundles are cobordant to flat bundles, addressing Haefliger's question.

Findings

Powers of Euler and Pontryagin classes are non-trivial in the cohomology of classifying spaces.

Certain sphere bundles are cobordant to flat bundles, linking their invariants.

Corrects a previous claim about classes in the smooth group cohomology of $ ext{Diff}_+( ext{S}^3)$.

Abstract

Morita showed that for each power of the Euler class, there are examples of flat $\mathbb{S}^1$-bundles for which the power of the Euler class does not vanish. Haefliger asked if the same holds for flat odd-dimensional sphere bundles. In this paper, for a manifold $M$ with a free torus action, we prove that certain $M$-bundles are cobordant to a flat $M$-bundle and as a consequence, we answer Haefliger's question. We show that the powers of the Euler class and Pontryagin classes $p_i$ for $i\leq n-1$ are all non-trivial in $H^*(\text{BDiff}^{\delta}_+(\mathbb{S}^{2n-1});\mathbb{Q})$. In the appendix, Nils Prigge corrects a claim by Haefliger about the vanishing of certain classes in the smooth group cohomology of $\text{Diff}_+(\mathbb{S}^3)$.