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Higher Equivariant Excision
Algebraic Topology · Submitted 7 Jul 2015
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We develop a theory of Goodwillie calculus for functors between $G$-equivariant homotopy theories, where $G$ is a finite group. | contribution 1.00 | 0.96 | 0.12 | asserted 1.00 |
| s1 | We construct $J$-excisive approximations of a homotopy functor for any finite $G$-set $J$. | contribution 0.96 | 0.95 | 0.10 | asserted 1.00 |
| s2 | These fit together into a poset, the Goodwillie tree, that extends the classical Goodwillie tower. | contribution 0.80 | 0.86 | 0.10 | asserted 1.00 |
| s3 | 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. | result 0.89 | 0.94 | 0.36 | asserted 1.00 |
| s4 | 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. | contribution 0.56 | 0.93 | 0.17 | asserted 1.00 |
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The paper
Original 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.
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What this paper does with the arXiv papers it cites
What it cites · 16 sentences without a confident label
Builds on 1 paper
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These results generalize immediately to the indexed power functors on $G$-spectra $(-)^∧K_O^ G→_O^G$ of [ref] and their family versions, but it is the functors from pointed spaces that will play a role in the calculation of the derivatives of the identity of [ref].
On the non-existence of elements of Kervaire invariant one · The genuine tower of the stable indexed powers · 0.80
Uses the method or tool of 1 paper
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We use the technique of [ref] and [ref] of factorizing the map $t_JΦΦ(X)→T_JΦ(X)$ through a certain $J_+|_H$-cube $Y∈𝒟^𝒫(J_+|_H)_a$.
A streamlined proof of Goodwillie's n-excisive approximation · Equivariant excisive approximations · 1.00
Cites as support 1 paper
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At least when $2∈A$ is invertible, the functor $(A;M(-))$ is $ℤ/2$-excisive (see [ref] and [ref] ). * Let $K$ be a finite $G$-set and let $E$ be an orthogonal $G$-spectrum.
Stable real K-theory and real topological Hochschild homology · Higher equivariant excision · 0.98
Cites as background 2 papers
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We review some of the foundations of equivariant diagrams, recollected from [ref] and [ref] .
Equivariant diagrams of spaces · Preliminaries on equivariant diagrams · 0.99
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The space $Eℱ_n$ is a universal space for $(G,Σ_n)$-vector bundles, and it occurs in the theory of $N_∞$-operads of [ref] , in the equivariant Snaith-splitting of [ref] , and in the proof of Sullivan's conjecture of [ref] .
Operadic multiplications in equivariant spectra, norms, and transfers · Higher equivariant excision · 0.98
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