In this paper, we examine the interaction of $\pi_1(M)$ for $C^\infty$ manifolds $M$ with Bott’s original obstacle to integrability, and Godbillon-Vey and Cheeger-Simons for foliate structures. We study interactions with differential geometric invariants such as invariants. We prove that the rings of the higher Pontrajagin classes and the higher Chern classes of the integrable subbundle $E$ of the tangent bundle of manifolds vanish above the dimension $2k$ of $k=dim(TM/E)$. increase. Chern rings are rings generated by $i^*y \cup p_j(TM/E)$ and $i^*y \cup c_j(TM/E)$ respectively, where $p_j$ is the $j$ Pontrjagin class. , $c_j$ $j$-th Chern class, $i:M \to B\pi$ and $\pi=\pi_1(BG)$, $BG$ is the classification space E$ of the holonomy groupoid corresponding to $ and $y \in H^*(B\pi)$ provided that the fundamental group of $BG$ satisfies the Novikov conjecture. In addition, we show vanishing higher Pontrjagin and Chern rings produced by $i^*x \cup p_j(TM/E)$ and $i^*x \cup c_j(TM/E)$ , whereas $(M,\mathcal{F})$ provided by $i:M \to BG$, $BG$ and $x \in H^*(BG)$ is the tangent line The leaf whose bundle is $E$. We give examples of this disorder, as well as higher Godbillon-Bay and Zieger-Simmons invariants.
