On invariants of foliated sphere bundles

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 $i\le n-1$ are all non-trivial in $H^{\ast }({\text{BDiff}}_{+}^{\delta }({\mathbb{S}}^{2n-1});\mathbb{Q})$. In the appendix, Nils Prigge corrects a claim by Haefliger about the vanishing of certain classes in the smooth group cohomology of ${\text{Diff}}_{+}({\mathbb{S}}^3)$.

13 pages, the appendix by Nils Prigge in the previous version will appear separately. This version will appear in Commentarii Mathematici Helvetici