geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Be?linson-Bernstein localization?
The generalization of the concept of homotopy group from homotopy theory and stable homotopy theory to equivariant homotopy theory and equivariant stable homotopy theory.
For a pointed topological G-space and a closed subgroup, the th unstable -equivariant homotopy group of is simply the ordinary -th homotopy group of the -fixed point space :
With denoting the quotient space, this is equivalently the -homotopy classes of -equivariant continuous functions from the smash product to :
In this form the definition directly generalizes to G-spectra and hence to stable equivariant homotopy groups: for a G-spectrum, and , write
where now is the -fold suspension of the suspension spectrum of the n-sphere and now denotes the hom functor in the equivariant stable homotopy category..
In particular, the (∞,1)-category of -spectra is stable, and hence it is cotensored over the (∞,1)-category of spectra; using this structure, we produce a chain of natural equivalences
where is the -fixed point spectrum of (e.g. Schwede 15, prop. 7.2).
The identification via fixed point spectra in turn is the equivariant cohomology of the point,
due to the base change adjunction
If is a real orthogonal -representation and is a subgroup, then we define the functor by
for fixed these are functorial in , and the resulting structure is called a -Mackey functor by Lewis.
Similarly, if denotes the real orthogonal representation ring and is a real orthogonal virtual representation of , then we define the functor by
These together bear the structure of an -graded Mackey functor, i.e. they take the form of a functor
Often one fixes , in which case one simply writes
See more at C_2-equivariant homotopy groups of spheres.
Consider genuine G-spectra modeled on a G-universe .
For a finite based G-CW complex and base topological G-space , write
for the colimit over -homotopy classes of maps between suspensions , where runs through the indexing spaces in the universe and denotes its representation sphere.
The equivariant stable homotopy groups of are
And for subgroups
Let be a finite group. For a -equivariant spectrum modeled as an orthogonal spectrum with -action, then for the th equivariant homotopy group of is the colimit
where
denotes the fundamental representation of
;
is its representation sphere;
is the value of in RO(G)-degree ;
is the set of homotopy classes of -equivariant maps of pointed topological G-spaces.
(e.g. Schwede 15, section 3)
More generally for a subgroup then one writes for the -equivariant subgroups of with regarded now as an -equivariant spectrum, via restriction of the action.
(e.g. Schwede 15, p. 16)
For a pointed topological G-space, then by the discussion there) the formula for the equivariant homotopy groups of its equivariant suspension spectrum reduces to
which in turn decomposes as a direct sum of ordinary homotopy groups of Weyl group-homotopy quotients of naive fixed point spaces – see at tom Dieck splitting.
For the equivariant sphere spectrum the tom Dieck splitting gives that its 0th equivariant homotopy group is the free abelian group on the set of conjugacy classes of subgroups of :
(e.g. Schwede 15, p. 64)
In the case , the RO(G)-graded stable equivariant homotopy groups of the sphere have been determined in the following ranges:
stems and coweights (Guillou-Isaksen 24);
and coweights (Guillou-Isaksen 24).
As -varies over the subgroups of a -equivariant spectrum , the -equivariant homotopy groups organize into a contravariant additive functor from the full subcategory of the equivariant stable homotopy category (called a Mackey functor)
(e.g. Schwede 15, p. 16 and section 4)
(…)
See at equivariant Whitehead theorem.
Gaunce Lewis, The equivariant Hurewicz map (1992) (TAMS)
John Greenlees, Peter May, Equivariant stable homotopy theory, in I.M. James (ed.), Handbook of Algebraic Topology , pp. 279-325. 1995. (pdf)
Peter May, section IX.4 of 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. With contributions by M. Cole, G. Comeza˜na, S. Costenoble, A. D. Elmenddorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. (pdf)
Stefan Schwede, Lectures on Equivariant Stable Homotopy Theory, 2015 (pdf)
Bertrand Guillou, Daniel Isaksen, -Equivariant Stable Stemsm (arXiv:2404.14627)
Last revised on April 24, 2024 at 18:39:51. See the history of this page for a list of all contributions to it.