On invariants of foliated sphere bundles

Abstract

Morita [Osaka J. Math. 21 (1984), 545–563] showed that for each integer $k\ge 1$, there are examples of flat ${\mathbb{S}}^1$-bundles for which the $k$-th power of the Euler class does not vanish. Haefliger [Enseign. Math. (2) 24 (1978), 154] 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 all monomials in the Euler class and Pontryagin classes $p_i$ for $i\le n-1$ are non-trivial in $H^{\ast }({\mathrm{BDiff}}_{+}^{\delta }({\mathbb{S}}^{2n-1});\mathbb{Q})$.