The bordism version of the h-principle

TL;DR

This paper proves the bordism version of the h-principle, establishing the equivalence of certain cohomology theories for open stable differential relations, enabling explicit computations of invariants of solutions using stable homotopy theory.

Contribution

It demonstrates the bordism version of the h-principle for a broad class of differential relations and determines the homotopy type of associated cohomology theories, extending classical results.

Findings

Cohomology theories $k^*$ and $h^*$ are equivalent for many differential relations.

The homotopy type of $h^*$ is explicitly determined.

Application to classical theorems like Pontrjagin-Thom and Barratt-Priddy-Quillen.

Abstract

In view of the Segal construction each category with a coherent operation gives rise to a cohomology theory. Similarly each open stable differential relation $R$ imposed on smooth maps of manifolds determines cohomology theories $k^*$ and $h^*$; the cohomology theory $k^*$ describes invariants of solutions of $R$, while $h^*$ describes invariants of so-called stable formal solutions of $R$. We prove the bordism version of the h-principle: The cohomology theories $k^*$ and $h^*$ are equivalent for a fairly arbitrary open stable differential relation $R$. Furthermore, we determine the homotopy type of $h^*$. Thus, we show that for a fairly arbitrary open stable differential relation $R$, the machinery of stable homotopy theory can be applied to perform explicit computations and determine invariants of solutions. In the case of the differential relation whose solutions are all maps, our construction amounts to the Pontrjagin-Thom construction. In the case of the covering differential relation our result is equivalent to the Barratt-Priddy-Quillen theorem asserting that the direct limit of classifying spaces $B\Sigma_n$ of permutation groups $\Sigma_n$ of finite sets of n elements is homology equivalent to each path component of the infinite loop space $\Omega^{\infty}S^{\infty}$. In the case of the submersion differential relation imposed on maps of dimension $d=2$ the cohomology theories $k^*$ and $h^*$ are not equivalent. Nevertheless, our methods still apply and can be used to recover the Madsen-Weiss theorem (the Mumford Conjecture).