nLab equivariant rationalization

Definition

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

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

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

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

to a rational $G$-space $L_{\mathbb{Q}}X$ which induces isomorphisms on all rationalized equivariant homotopy groups:

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

Properties

Via Elmendorf’s theorem

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

Related concepts

• equivariant Chern character

• equivariant Chern-Dold character

• rational equivariant homotopy theory

• rational stable equivariant homotopy theory

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)