Equivariant Trees and Partition Complexes

TL;DR

This paper introduces $G$-equivariant partitions and trees, demonstrating their $G$-homotopy equivalence, and develops equivariant versions of Quillen's Theorems A and B, advancing the understanding of equivariant homotopy theory.

Contribution

It defines $G$-equivariant partitions and trees, proving their $G$-homotopy equivalence and extending classical theorems to the equivariant setting.

Findings

$G$-equivariant partitions form complexes with $G$-homotopy equivalence to $G$-trees

Develops equivariant versions of Quillen's Theorems A and B

Generalizes non-equivariant results to the equivariant context

Abstract

We introduce two definitions of $G$-equivariant partitions of a finite $G$-set, both of which yield $G$-equivariant partition complexes. By considering suitable notions of equivariant trees, we show that $G$-equivariant partitions and $G$-trees are $G$-homotopy equivalent, generalizing existing results for the non-equivariant setting. Along the way, we develop equivariant versions of Quillen's Theorems A and B, which are of independent interest.