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

TL;DR

This paper explores the relationship between foliation invariants, Bott's integrability obstruction, and higher characteristic classes, establishing vanishing results under certain conjectural conditions and providing examples of these invariants.

Contribution

It proves vanishing theorems for higher Pontrjagin and Chern rings related to integrable subbundles under Novikov conjecture assumptions and connects these to higher Godbillon-Vey and Cheeger-Simons invariants.

Findings

Higher Pontrjagin and Chern rings vanish above dimension 2k under Novikov conjecture.

Vanishing of higher characteristic rings generated by specific cohomology classes.

Examples illustrating obstructions and higher invariants in foliation theory.

Abstract

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.