The weak b-principle: Mumford conjecture

TL;DR

This paper introduces oriented colored broken submersions, a new class of maps satisfying a b-principle variant, and demonstrates how the Madsen-Weiss theorem fits into this framework, linking it to stability theorems.

Contribution

It defines a new class of maps called oriented colored broken submersions and shows their relation to the Mumford conjecture via the b-principle and stability theorems.

Findings

Oriented colored broken submersions approximate submersions in dimension 2.

The b-principle applies to the Madsen-Weiss theorem.

Stability theorems imply the Mumford conjecture.

Abstract

In this note we introduce and study a new class of maps called oriented colored broken submersions. This is the simplest class of maps that satisfies a version of the b-principle and in dimension 2 approximates the class of oriented submersions well in the sense that every oriented colored broken submersion of dimension 2 to a closed simply connected manifold is bordant to a submersion. We show that the Madsen-Weiss theorem (the standard Mumford Conjecture) fits a general setting of the b-principle. Namely, a version of the b-principle for oriented colored broken submersions together with the Harer stability theorem and Miller-Morita theorem implies the Madsen-Weiss theorem.