Higher Equivariant Excision

TL;DR

This paper extends Goodwillie calculus to $G$-equivariant homotopy theories, constructing a new Goodwillie tree framework that generalizes classical towers and reveals new splitting phenomena in equivariant contexts.

Contribution

It develops a theory of $J$-excisive approximations for functors between $G$-equivariant homotopy theories, forming a comprehensive Goodwillie tree that generalizes classical results.

Findings

Constructed $J$-excisive approximations for $G$-equivariant functors.

Proved convergence of the Goodwillie tree for certain $G$-spaces.

Reinterpreted Tom Dieck-splitting as a special case of a general splitting phenomenon.

Abstract

We develop a theory of Goodwillie calculus for functors between $G$-equivariant homotopy theories, where $G$ is a finite group. We construct $J$-excisive approximations of a homotopy functor for any finite $G$-set $J$. These fit together into a poset, the Goodwillie tree, that extends the classical Goodwillie tower. We prove convergence results for the tree of a functor on pointed $G$-spaces that commutes with fixed-points, and we reinterpret the Tom Dieck-splitting as an instance of a more general splitting phenomenon that occurs for the fixed-points of the equivariant derivative of these functors. As our main example we describe the layers of the tree of the identity functor in terms of the equivariant Spanier-Whitehead duals of the partition complexes.