A degree theorem for the simplicial closure of Auter space

TL;DR

This paper extends the degree theorem for Auter space to its simplicial closure, showing that the subcomplex of graphs with degree at most d remains highly connected, thus broadening the understanding of its topological structure.

Contribution

It generalizes the degree theorem to the simplicial closure of Auter space, providing new insights into its connectivity properties.

Findings

The subcomplex of graphs with degree ≤ d is (d-1)-connected.

The degree concept is extended to the simplicial closure of Auter space.

The connectivity results mirror those of the original degree theorem.

Abstract

The degree of a based graph is the number of essential nonbasepoint vertices after generic perturbation. Hatcher--Vogtmann's degree theorem states that the subcomplex of Auter space of graphs of degree at most d is (d-1)-connected. We extend the definition of degree to the simplicial closure of Auter space and prove a version of Hatcher--Vogtmann's result in this context.