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?
Given a -equivariant spectrum, its fixed point spectrum is the plain spectrum which is the value of the derived functor of the naive -fixed point functor on . This constitutes a suitably derived functor
(e.g. Schwede 15, def. 7.1, Mandell-May 02, arojnd thm. 3.3)
More generally, for a normal subgroup, there is the -fixed point spectrum regarded as -spectrum, equivariant under the remaining quotient group :
(Mandell-May 02, def. 3.8, Greenlees-Shipley 11, Prop. 3.3)
These are derived right adjoint to the operation of regarding a -spectrum as a -spectrum, via the projection
(Mandell-May 02, theorem 3.12, Greenlees-Shipley 11, Prop. 3.3)
The plain homotopy groups of the -fixed point spectrum of are the equivariant homotopy groups of the -spectrum :
(e.g. Schwede 15, prop. 7.2)
For an equivariant suspension spectrum the fixed point spectrum is given by the tom Dieck splitting formula
(e.g. Schwede 15, example 7.7)
categorical fixed point spectrum?
L. Gaunce Lewis, Jr., Splitting theorems for certain equivariant spectra, Memoirs of the AMS, number 686, March 2000, Volume 144 (pdf)
Michael Mandell, Peter May, Equivariant orthogonal spectra and S-modules, Mem. Amer. Math. Soc. 159 (2002), no. 755, x+108. MR 2003i:55012 (pdf, K-theory archive)
John Greenlees, Brooke Shipley, p. 18 of An algebraic model for rational torus-equivariant spectra (arXiv:1101.2511)
John Greenlees, Brooke Shipley, Fixed point adjunctions for equivariant module spectra, Algebr. Geom. Topol. 14 (2014) 1779-1799 (arXiv:1301.5869)
Stefan Schwede, Lectures on Equivariant Stable Homotopy Theory, 2015 (pdf)
Last revised on December 30, 2018 at 18:27:21. See the history of this page for a list of all contributions to it.