nLab G-spectrum

Idea

For $G$ a compact Lie group (or more generally a compact topological group) the concept of $G$-spectrum (or $G$-equivariant spectrum) is the generalization of that of spectrum as one passes from stable homotopy theory to equivariant stable homotopy theory, or more generally, as $G$ is allow to vary, to global equivariant stable homotopy theory.

Where the ordinary concept of spectrum is given in terms of looping and delooping of ordinary topological spaces by ordinary spheres, a $G$-spectrum is instead given by looping and delooping of topological G-spaces with respect to representation spheres of $G$, namely one-point compactifications of linear $G$-representations, for all representations appearing in a chosen “G-universe”.

Such a G-universe is called complete if it contains every irreducible representation of $G$, and the spectra modeled on such a complete $G$-universe are the genuine $G$-spectra. At the other extreme, if the G-universe contains only the trivial representations, then the resulting spectra are the spectra with G-action, also called naive G-spectra for emphasis of the distinction to the previous case.

The genuine $G$-spectra are richer than spectra with G-action and have better homotopy-theoretic properties. In particular the genuine equivariant cohomology theories which they represent have suspension isomorphisms for suspension by all representation spheres and with respect to RO(G)-grading.

When $G$ is the trivial group, a $G$-spectrum is also known as a coordinate-free spectrum.

Models

There are various equivalent ways to present genuine $G$-spectra.

Indexed on all representations

(May 96, chapter XII, Greenlees-May, section 2)

Fix a G-universe. For $V$ any orthogonal representation in the universe, write $S^V$ for its representation sphere. For $V \hookrightarrow W$ a subrepresentation, write $W-V$ for the orthogonal complement representation.

A $G$-prespectrum $E$ is an assignment of a pointed G-space $E_V$ to each $G$-representation $V$ (in the given G-universe), equipped for each subrepresentation $V \hookrightarrow W$ with a pointed $G$-equivariant continuous function

$$\sigma_{V,W} \;\colon\; S^{W-V} \wedge E_V \longrightarrow E_W$$

such that

1. $\sigma_{V,V} = id$;

2. for any $V \hookrightarrow W \hookrightarrow Z$ we have commuting diagrams

   $$\array{ S^{Z-W}\wedge S^{W-V} \wedge E_V &\stackrel{S^{Z-W}\wedge(\sigma_{V,W})}{\longrightarrow}& S^{Z-W} \wedge E_W \\ \downarrow && \downarrow^{\mathrlap{\sigma_{W,Z}}} \\ S^{Z-V} \wedge E_V &\stackrel{\sigma_{V,Z}}{\longrightarrow}& E_Z }$$

Write

$$\tilde \sigma_{V,W} \;\colon\; E_V \longrightarrow \Omega^{W-V}E_W$$

for the adjuncts of these structure maps.

A $G$-prespectrum is called (at least in (May 96, chapter XII))

• a $G$-$\Omega$-spectrum if all the $\sigma_{V,W}$ are weak homotopy equivalences;

• a $G$-spectrum if all the $\sigma_{V,W}$ are homeomorphisms.

Via orthogonal spectra with $G$-action

While the definition of spectra indexed on all representations manifestly relates to the suspension isomorphism for smashing with representation spheres and shifting in RO(G)-grading, the information encoded in the objects in this definition has much redundancy. A “smaller” definition of genuine $G$-spectra is given by orthogonal spectra equipped with $G$-action (Mandell-May 04, Schwede 15).

Via Mackey functors

For $G$ a finite group then genuine $G$-spectra are equivalent to Mackey functors on the category of finite G-sets.

(Guillou-May 11, theorem 0.1, Barwick 14, below example B.6).

Via excisive functors

Characterization of $G$-spectra via excisive functors on G-spaces is in (Blumberg 05).

Properties

Relation to spectra with $G$-action

(e.g. Carlsson 92, p. 14, GreenleesMay, p.16)

Relation to Borel-equivariant spectra

In the general context of (global) equivariant stable homotopy theory, Borel-equivariant spectra are those which are right induced from plain spectra, hence which are in the essential image of the right adjoint to the forgetful functor from equivariant spectra to plain spectra.

(Schwede 18, Example 4.5.19)

Equivariant stable Whitehead theorem

The equivariant version of the stable Whitehead theorem holds:

a map of $G$-spectra $f \colon E \longrightarrow F$ is a weak equivalence (e.g. an $RO(G)$-degree-wise weak homotopy equivalence of topological G-spaces in the model via indexing on all representations) precisely it if induces isomorphisms $\pi_\bullet(f) \colon \pi_\bullet(E) \longrightarrow \pi_{\bullet}(F)$ on all equivariant homotopy group Mackey functors.

(Greenlees-May, theorem 2.3)

Related concepts

• equivariant stable homotopy category

• RO(G)-grading

• fixed point spectrum

• geometric fixed points

Examples

• equivariant sphere spectrum

References

Good lecture notes are

• Andrew Blumberg, Equivariant homotopy theory, 2017 (pdf, GitHub)

The concept of genuine $G$-spectra is due to

• L. Gaunce Lewis, Jr., Peter May, M. Steinberger, and J. E. McClure, Equivariant stable homotopy theory, Lecture Notes in Mathematics, vol. 1213, Springer-Verlag, Berlin, 1986, With contributions by J. E. McClure. MR 866482 (88e:55002)

and in terms of orthogonal spectra due to

• Michael Mandell, Peter May, Equivariant orthogonal spectra and S-modules. Preprint, April 29, 2000, (KTheory:0408)

Further accounts:

• Tammo tom Dieck, Section II.6 of: Transformation Groups, de Gruyter 1987 (doi:10.1515/9783110858372)

and in the context of the Arf-Kervaire invariant problem:

• Michael Hill, Michael Hopkins, Doug Ravenel, The Arf-Kervaire invariant problem in algebraic topology: introduction, Current developments in mathematics, 2009, Int. Press, Somerville, MA, 2010, pp. 23–57. MR 2757358

  On the non-existence of elements of Kervaire invariant one, (arXiv:0908.3724)

  The Arf-Kervaire problem in algebraic topology: sketch of the proof, Current developments in mathematics, 2010, Int. Press, Somerville, MA, 2011, pp. 1–43

Surveys and introductions include

• Gunnar Carlsson, A survey of equivariant stable homotopy theory,Topology, Vol 31, No. 1, pp. 1-27, 1992 (pdf)

• John Greenlees, Peter May, section 2 of Equivariant stable homotopy theory, in I.M. James (ed.), Handbook of Algebraic Topology , pp. 279-325. 1995. (pdf)

Lecture notes include

• Peter May, Equivariant homotopy and cohomology theory CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996. (pdf)

• Stefan Schwede, Lectures on Equivariant Stable Homotopy Theory

See also

• Stefan Schwede, Global homotopy theory 2018 (arXiv:1802.09382)

Relation to Mackey functors:

• Bert Guillou, Peter May, Models of $G$-spectra as presheaves of spectra, (arXiv:1110.3571)

  Permutative $G$-categories in equivariant infinite loop space theory (arXiv:1207.3459)

• Denis Nardin, section 2.6 of Stability and distributivity over orbital ∞-categories, 2012 (pdf)

• Clark Barwick, Spectral Mackey functors and equivariant algebraic K-theory (I), Adv. Math., 304:646–727 (arXiv:1404.0108)

For more references see at equivariant stable homotopy theory and at Mackey functor

Characterization via excisive functors is in

• Andrew Blumberg, Continuous functors as a model for the equivariant stable homotopy category (arXiv:math.AT/0505512)

In the case of a cyclic group of prime order, genuine $G$-spectra admit a simple model which amounts to specifying a spectrum $E$, a $G$-action on $E$, a genuine fixed point spectrum $E^G$, and a diagram $E_{hG} \to E^G \to E^{hG}$. See Example 3.29 in:

• Saul Glasman, Stratified categories, geometric fixed points and a generalized Arone-Ching theorem, arXiv:1507.01976.

A perspective on the category of genuine G-spectra as a lax limit over those of “naive” G-spectra is given in

• David Ayala, Aaron Mazel-Gee, Nick Rozenblyum, A naive approach to genuine $G$-spectra and cyclotomic spectra (arXiv:1710.06416)

Universal properties of the (∞,1)-category of genuine $G$-spectra are considered in

• Bastiaan Cnossen, Twisted ambidexterity in equivariant homotopy theory, arXiv.

• Tim Campion, Free duals and a new universal property for stable equivariant homotopy theory, arXiv.