Bott Integrability and Higher Integrability; Higher Cheeger-Simons and Godbillon-Vey Invariants

This paper studies the interaction of $\pi_1(M)$ for a $C^\infty$ manifold $M$ with Bott's original obstruction to integrability, and with differential geometric invariants such as Godbillon-Vey and Cheeger-Simons invariants of a foliation. We prove that the ring of higher Pontrjagin and higher Chern classes of an integrable subbundle $E$ of the tangent bundle of a manifold vanishes above dimension $2k$ where $k=dim(TM/E)$, and where the higher Pontrjagin and Chern rings are rings generated by $i^*y \cup p_j(TM/E)$ and by $i^*y \cup c_j(TM/E)$ respectively, with $p_j$ the $j$-th Pontrjagin class, $c_j$ the $j$-th Chern class, $i:M \to B\pi$ and $\pi=\pi_1(BG)$, where $BG$ is the classifying space of the holonomy groupoid corresponding to $E$ and $y \in H^*(B\pi)$, provided that the fundamental group of $BG$ satisfies the Novikov conjecture. In addition, we show the vanishing of higher Pontrjagin and Chern rings generated by $i^*x \cup p_j(TM/E)$, and by $i^*x \cup c_j(TM/E)$ as before but with $i:M \to BG$, $BG$ as above and $x \in H^*(BG)$ provided $(M,\mathcal{F})$ satisfied the foliated Novikov conjecture, where $\mathcal{F}$ is the foliation whose tangent bundle is $E$. We give examples of this obstruction and of higher Godbillon-Vey and Cheeger-Simons invariants.