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G-spectrum

Contents

Context

Stable Homotopy theory

Representation theory

Contents

Idea

For GG a compact Lie group (or more generally a compact topological group) the concept of GG-spectrum (or GG-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 GG 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 GG-spectrum is instead given by looping and delooping of topological G-spaces with respect to representation spheres of GG, namely one-point compactifications of linear GG-representations, for all representations appearing in a chosen “G-universe”.

Such a G-universe is called complete if it contains every irreducible representation of GG, and the spectra modeled on such a complete GG-universe are the genuine GG-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 GG-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 GG is the trivial group, a GG-spectrum is also known as a coordinate-free spectrum.

Models

There are various equivalent ways to present genuine GG-spectra.

Indexed on all representations

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

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

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

σ V,W:S WVE VE W \sigma_{V,W} \;\colon\; S^{W-V} \wedge E_V \longrightarrow E_W

such that

  1. Σ V,V=id\Sigma_{V,V} = id;

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

    S ZWS VWE V S ZW(σ V,W) S ZWE W Σ Z,W S ZVE V σ V,Z E Z \array{ S^{Z-W}\wedge S^{V-W} \wedge E_V &\stackrel{S^{Z-W}\wedge(\sigma_{V,W})}{\longrightarrow}& S^{Z-W}E_W \\ \downarrow && \downarrow^{\mathrlap{\Sigma_{Z,W}}} \\ S^{Z-V}E_V &\stackrel{\sigma_{V,Z}}{\longrightarrow}& E_Z }

Write

σ˜ V,W:E VΩ WVE W \tilde \sigma_{V,W} \;\colon\; E_V \longrightarrow \Omega^{W-V}E_W

for the adjuncts of these structure maps.

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

Via orthogonal spectra with GG-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 GG-spectra is given by orthogonal spectra equipped with GG-action (Mandell-May 04, Schwede 15).

Via Mackey functors

For GG a finite group then genuine GG-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 GG-spectra via excisive functors on G-spaces is in (Blumberg 05).

Properties

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

(Greenlees-May, theorem 2.3)

Examples

References

Good lecture notes are

The concept of genuine GG-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

also (developed for 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

Lecture notes include

See also

Relation to Mackey functors:

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

Characterization via excisive functors is in

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

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

Last revised on October 24, 2018 at 14:31:41. See the history of this page for a list of all contributions to it.