A combinatorial proof of the Degree Theorem in Auter space

TL;DR

This paper presents a new local proof of the Degree Theorem in Auter space using discrete Morse theory, which simplifies understanding and potentially extends to broader contexts in algebraic topology.

Contribution

It introduces a local, Morse-theoretic proof of the Degree Theorem, contrasting with the traditional global approach, facilitating generalizations.

Findings

A new proof of the Degree Theorem using discrete Morse theory

The filtration of Auter space into subspaces A_{n,k} is (k-1)-connected

The proof approach is more adaptable for future generalizations

Abstract

We use discrete Morse theory to give a new proof of the Degree Theorem in Auter space A_n. There is a filtration of A_n into subspaces A_{n,k} using the degree of a graph, and the Degree Theorem says that each A_{n,k} is (k-1)-connected. This result is useful, for example to calculate stability bounds for the homology of Aut(F_n). The standard proof of the Degree Theorem is global in nature. Here we give a proof that only uses local considerations, and lends itself more readily to generalization.