Equivariant Stable Homotopy Theory: Overview

This file is the index for the concepts/equivariant-stable-homotopy/ folder. It lists planned and written subtopic notes, organizes them by theme, and collects the canonical references for the field. Use it to decide what to write next without needing to re-survey the landscape.

Notes in This Folder

File Status Topic
g-spaces-and-equivariant-maps.md ✅ Written G-spaces, orbit category, naive vs. genuine, G-CW complexes, Elmendorf’s theorem
equivariant-postnikov-and-slice.md ✅ Written Equivariant Postnikov towers (unstable), the slice filtration, slice spectral sequence
bredon-cohomology.md 🔲 Planned Coefficient systems, chain complex of coefficient systems, Bredon (co)homology, computations
mackey-functors.md ✅ Written Mackey functors, the Burnside category, spectral Mackey functors (Barwick)
g-spectra.md ✅ Written Genuine G-spectra indexed on a complete universe, orthogonal G-spectra, naive vs. genuine stable
ro-g-graded-cohomology.md 🔲 Planned RO(G)-graded homotopy groups, representation spheres, RO(G)-graded Bredon cohomology
fixed-point-spectra.md 🔲 Planned Geometric fixed points \(\Phi^H\), categorical fixed points \((-)^H\), homotopy fixed points \((-)^{hH}\); the Tate diagram
wirthmuller-and-adams.md 🔲 Planned Wirthmuller isomorphism \(i_! \simeq i_*\), Adams isomorphism, formal duality
norm-maps.md 🔲 Planned Multiplicative norm \(N_H^G\), \(N_\infty\)-operads, commutative equivariant ring spectra
tate-spectra.md 🔲 Planned Tate construction \(X^{tG}\), Tate diagonal, generalized Tate cohomology, TC and THH
equivariant-k-theory.md 🔲 Planned Segal’s \(K_G(X)\), KR-theory (Atiyah), Bott periodicity, Atiyah-Segal completion theorem
hhr-theorem.md 🔲 Planned Kervaire invariant one problem, \(MU^{((C_8))}\), Detection/Periodicity/Gap theorems
global-equivariant-homotopy.md 🔲 Planned Global spectra (Schwede), simultaneous \(G\)-actions for all compact Lie groups \(G\)
parametrized-spectra.md 🔲 Planned Parametrized \(\infty\)-categories (Barwick et al.), \(G\)-\(\infty\)-categories, equivariant \(\infty\)-operads

Subtopic Map

Unstable Equivariant Homotopy Theory

Subtopic Key Idea Primary Source
G-spaces and Elmendorf Genuine G-spaces ≃ presheaves on \(\mathcal{O}_G\) Blumberg §1, Elmendorf 1983
Bredon cohomology Cohomology with coefficient systems; k-invariants for equivariant Postnikov towers Bredon 1967, Blumberg §1.4
Equivariant Postnikov towers Objectwise via Elmendorf; fibers are \(K(\underline{M}, n)\); k-invariants in Bredon cohomology Blumberg §1.3 remark, May Alaska §3
Smith theory Fixed-point sets of \(\mathbb{Z}/p\)-actions on mod-\(p\) homology spheres May Alaska §3

Equivariant Stable Homotopy Theory

Subtopic Key Idea Primary Source
Mackey functors Abelian-group-valued functors with restriction + transfer on \(\mathcal{O}_G\); coefficient objects for RO(G)-theories Dress 1973, Webb 2000, Blumberg §3
Genuine G-spectra Spectra indexed on a complete \(G\)-universe \(\mathcal{U}\); two model structures (naive vs. genuine) are now different categories Lewis-May-Steinberger 1986, Blumberg §2
RO(G)-graded cohomology Suspension by representation spheres \(S^V\); homotopy groups \(\pi_V^H(X) = [S^V, X]^H\) May Alaska §XIII–XIV, Costenoble-Waner
Fixed-point spectra Three flavors: \(X^H\) (categorical), \(\Phi^H X\) (geometric), \(X^{hH}\) (homotopy); Tate sequence \(X^H \to X^{hH} \to X^{tH}\) Greenlees-May 1995, Blumberg §2.4
Wirthmuller isomorphism \(i_! \simeq i_* \otimes \text{(representation twist)}\) for restriction \(i^*: \text{Sp}^G \to \text{Sp}^H\); equivariant Poincaré duality May 2003, Fausk-Hu-May 2003
Adams isomorphism \((G_+ \wedge_H X)^G \simeq X^H\) for a free \(H\)-spectrum \(X\); relates orbits and fixed points Haugseng et al. 2016, Blumberg §2.4
Norm maps \(N_H^G: \text{Sp}^H \to \text{Sp}^G\); multiplicative transfer; \(N_\infty\)-operads; no unstable analogue HHR 2016, Blumberg-Hill 2015
Tate spectra \(X^{tG} = \widetilde{EG} \wedge X^{hG}\); Tate diagonal \(\Delta_p: X \to (X^{\otimes p})^{tC_p}\); fundamental for TC Greenlees-May 1995, Nikolaus-Scholze 2018
Slice filtration Equivariant analogue of Postnikov tower for G-spectra; slice cells \(G_+ \wedge_H S^{m\rho_H}\); indexed by representation dimension HHR 2016, Hill 2011, Blumberg §5.2

Applications

Subtopic Key Idea Primary Source
KR-theory Spaces with involution; real \(K\)-theory with 8-fold periodicity; \(K_G(X)\) for general \(G\) Atiyah 1966, Segal 1968
HHR theorem \(\theta_j \notin \pi_{2^{j+1}-2}^s\) for \(j \geq 7\); uses genuine \(C_8\)-spectra, slice SS, norm \(N_{C_2}^{C_8}(MU_\mathbb{R})\) HHR 2009/2016, Miller 2011
Global equivariant homotopy Uniform \(G\)-equivariance for all compact Lie \(G\) simultaneously; ultra-commutative ring spectra Schwede 2018
Parametrized / \(\infty\)-categorical \(G\)-\(\infty\)-categories, Barwick’s effective Burnside \(\infty\)-category, spectral Mackey functors Barwick 2014, Nardin-Shah 2022

Dependency Graph

flowchart TD
    A["G-spaces & Elmendorf\ng-spaces-and-equivariant-maps.md ✅"]
    B["Bredon cohomology\nbredon-cohomology.md"]
    C["Equivariant Postnikov & Slice\nequivariant-postnikov-and-slice.md"]
    D["Mackey functors\nmackey-functors.md"]
    E["Genuine G-spectra\ng-spectra.md"]
    F["RO(G)-graded cohomology\nro-g-graded-cohomology.md"]
    G["Fixed-point spectra\nfixed-point-spectra.md"]
    H["Wirthmuller & Adams\nwirthmuller-and-adams.md"]
    I["Norm maps\nnorm-maps.md"]
    J["Tate spectra\ntate-spectra.md"]
    K["HHR theorem\nhhr-theorem.md"]
    L["KR-theory\nequivariant-k-theory.md"]

    A --> B
    A --> C
    B --> C
    A --> D
    D --> E
    E --> F
    E --> G
    G --> H
    G --> J
    E --> I
    F --> K
    I --> K
    C --> K
    G --> L
    F --> L

Master References

Reference Authors Year What It Covers Link
M392C Lecture Notes Blumberg (notes by Debray) 2017 G-spaces, Elmendorf, genuine G-spectra, Mackey functors, RO(G), HHR slice SS — the single best survey PDF
Equivariant Homotopy and Cohomology Theory May et al. 1996 Complete classical treatment: G-CW, RO(G), Wirthmuller, Adams, Tate; CBMS Alaska notes PDF
Equivariant Stable Homotopy Theory Lewis, May, Steinberger 1986 Definitive reference for genuine G-spectra on a complete universe; smash product, change-of-universe Springer LNM 1213
Equivariant Stable Homotopy Theory and the Kervaire Invariant Problem Hill, Hopkins, Ravenel 2021 Self-contained monograph: orthogonal G-spectra, slice filtration, HHR proof Cambridge
On the Non-Existence of Elements of Kervaire Invariant One Hill, Hopkins, Ravenel 2009/2016 Original HHR paper; norm maps, slice SS, Kervaire invariant one arXiv:0908.3724
Equivariant Stable Homotopy Theory Greenlees, May 1995 Handbook survey: all three fixed-point functors, Tate diagram, Wirthmuller/Adams PDF
Lectures on Equivariant Stable Homotopy Theory Schwede 2020 Clean modern treatment via orthogonal G-spectra; good complement to LMS PDF
A User’s Guide to G-Spectra Malkiewich draft 344pp draft: model structures on G-spectra, comparisons between flavors PDF
Basic Notions of Equivariant Stable Homotopy Theory Bohmann 2010 Short expository notes: three fixed-point functors, RO(G)-grading in a few pages PDF
Systems of Fixed Point Sets Elmendorf 1983 Original Elmendorf’s theorem via bar construction AMS
Equivariant Cohomology Theories Bredon 1967 Original definition of Bredon cohomology with coefficient systems Springer LNM 34
A Guide to Mackey Functors Webb 2000 Handbook survey: algebra of Mackey functors, resolutions, connections to representation theory PDF
Spectral Mackey Functors and Equivariant Algebraic K-Theory Barwick 2014 Mackey functors as excisive functors on the Burnside ∞-category; bridges classical and ∞-categorical arXiv:1404.0108
Prerequisites for Carlsson’s Lecture Adams 1984 Adams’s expository account of genuine G-spectra, representations, suspension spectra PDF
The Wirthmuller Isomorphism Revisited May 2003 Clean proof of Wirthmuller isomorphism; clarifies relation to Adams isomorphism arXiv:math/0206080
Isomorphisms Between Left and Right Adjoints Fausk, Hu, May 2003 Wirthmuller and Adams isomorphisms via abstract formal duality PDF
The Adams Isomorphism Revisited Kro 2024 Modern proof via equivariant semiadditivity in parametrized higher category theory arXiv:2311.04884
The Equivariant Slice Filtration: A Primer Hill 2011 Best entry point for the slice filtration before the full HHR paper arXiv:1107.3582
Generalized Tate Cohomology Greenlees, May 1995 Foundational AMS Memoir constructing Tate spectra for compact Lie groups PDF
On Topological Cyclic Homology Nikolaus, Scholze 2018 Tate diagonal \(\Delta_p\); reformulates TC via the Tate construction arXiv:1707.01799
Operadic Multiplications in Equivariant Spectra, Norms, and Transfers Blumberg, Hill 2015 \(N_\infty\)-operads; characterizes which norm maps a commutative equivariant ring spectrum admits arXiv:1309.1750
K-Theory and Reality Atiyah 1966 Original KR-theory; real \(K\)-theory for spaces with involution; 8-fold periodicity PDF
Equivariant K-Theory Segal 1968 Systematic \(K_G(X)\) for compact \(G\); equivariant Bott periodicity; localization theorem NUMDAM
Equivariant K-Theory and Completion Atiyah, Segal 1969 Atiyah-Segal completion theorem: \(K_G(X)^\wedge_{I(G)} \cong K(EG \times_G X)\) PDF
Global Homotopy Theory Schwede 2018 Definitive monograph on global equivariant homotopy via orthogonal spectra arXiv:1802.09382
Kervaire Invariant One [after HHR] Miller 2011 Bourbaki exposé surveying the HHR proof strategy; best orientation before the original arXiv:1104.4523
Parametrized Higher Category Theory and Higher Algebra Barwick et al. 2016 Foundations of parametrized \(\infty\)-categories for equivariant algebra arXiv:1608.03654
Parametrized and Equivariant Higher Algebra Nardin, Shah 2022 Parametrized \(\infty\)-operads; Day convolution; multiplicative equivariant structures arXiv:2203.00072