Madsen-Weiss for geometrically minded topologists

TL;DR

This paper provides a more geometrical proof of the Madsen-Weiss generalized Mumford conjecture, emphasizing geometric methods over homotopy theory, and introduces a geometric version of Harer stability involving folded maps.

Contribution

It offers an alternative, more geometrical proof of the Madsen-Weiss conjecture, utilizing a geometric version of Harer stability based on folded maps.

Findings

A geometric proof of the Madsen-Weiss conjecture is established.

A geometric version of Harer stability is formulated and proved.

The approach emphasizes geometric intuition over homotopy theoretical methods.

Abstract

We give an alternative proof of Madsen-Weiss' generalized Mumford conjecture. Our proof is based on ideas similar to Madsen-Weiss' original proof, but it is more geometrical and less homotopy theoretical in nature. At the heart of the argument is a geometric version of Harer stability, which we formulate as a theorem about folded maps.