nLab
equivariant rationalization

Contents

Context

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Representation theory

Rational homotopy theory

Contents

Definition

Let GG be a finite group (or more generally a compact Lie group).

Say that an equivariantly connected and nilpotent topological G-space YY (i.e. all fixed loci X HX^H are connected and nilpotent, for H clsdGH \subset_{clsd} G, with a common bound of nilpotency as HH ranges) is rational if all its equivariant homotopy groups π n(X H)\pi_n\big( X^H \big) (for nn \in \mathbb{N} and H cldGH \subset_{cld} G) admit the structure of rational vector spaces.

Given any topological G-space XX, a rationalization of XX is a morphism (a GG-equivariant continuous function)

Xη X L X X \overset{ \;\;\; \eta^{\mathbb{Q}}_X \;\;\; }{\longrightarrow} L_{\mathbb{Q}}X

to a rational GG-space L XL_{\mathbb{Q}}X which induces isomorphisms on all rationalized equivariant homotopy groups:

(η X (G/H)) *:π (X H)π ((L X) H). \big( \eta^{\mathbb{Q}}_X(G/H) \big)_\ast \;\colon\; \pi_\bullet( X^H ) \otimes \mathbb{Q} \overset{\simeq}{\longrightarrow} \pi_\bullet \Big( \big( L_{\mathbb{Q}}X \big)^H \Big) \,.

Properties

Via Elmendorf’s theorem

In other words, after regarding them, via Elmendorf's theorem, as (∞,1)-presheaves on the orbit category GOrbitsG Orbits of GG, the equivariant homotopy types of rational GG-spaces and their rationalizations are equivalently stage-wise over G/HGOrbitsG/H \in G Orbits plain rational spaces and rationalizations, respectively.

It follows from the fundamental theorem of dg-algebraic equivariant rational homotopy theory that, at least on equivariantly simply connected topological G-spaces, equivariant rationalization is given by the derived adjunction unit of the equivariant PL de Rham complex-Quillen adjunction.

References

  • Peter May, Section II.3 in: 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. Comezana, S. Costenoble, A. D. Elmenddorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. (ISBN: 978-0-8218-0319-6 pdf, pdf)

  • Georgia Triantafillou, Section 2.6 in Equivariant minimal models, Trans. Amer. Math. Soc. vol 274 pp 509-532 (1982) (jstor:1999119)

  • Laura Scull, p. 11 of: A model category structure for equivariant algebraic models, Transactions of the American Mathematical Society 360 (5), 2505-2525, 2008 (doi:10.1090/S0002-9947-07-04421-2)

Last revised on October 4, 2020 at 13:17:52. See the history of this page for a list of all contributions to it.