Equivariant Trees and Partition Complexes
Julia E. Bergner, Peter Bonventre, Maxine E. Calle, David Chan, Maru Sarazola
TL;DR
The paper develops two notions of G-equivariant partitions and corresponding G-trees, and proves that these two viewpoints are G-homotopy equivalent by equivariant analogues of Quillen's Theorems A and B. It builds two equivariant zig-zags linking G-partition complexes to G-trees, and analyzes the G-homotopy type of partition spaces by fixed-point data: contractible when partitions are not isovariant, and decomposed into Weyl-group indexed wedges of spheres when isovariant. A central result expresses the homology of spaces of G-trees in terms of equivariant Lie representations, generalizing the classical relation to the Lie operad. The work also establishes foundational tools for equivariant homotopy theory of partition complexes, with implications for operad theory and related algebraic structures in the presence of symmetry.
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.