On invariants of foliated sphere bundles

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 $\text{Diff}_+(\mathbb{S}^3)$.