nLab proper equivariant homotopy theory

Idea

Proper $G$-equivariant homotopy theory (DHLPS 19) is the variant of $G$-equivariant homotopy theory where the topological group $G$ is allowed to be non-compact (for instance a non-finite discrete group) but whose weak equivalences are still detected only on fixed point spaces of compact subgroups of $G$.

Hence in the presentation by topological G-spaces, proper $G$-equivariant homotopy theory is obtained by localizing at those equivariant continuous functions $f \;\colon\; X \to Y$ which induce weak homotopy equivalences $f^H \;\colon\; X^H \to Y^H$ for all compact subgroups $H \subset G$ (DHLPS 19, Def. 1.1.2).

Equivalently, under Elmendorf's theorem (DHLPS 19, p. 87) this is the homotopy theory of (∞,1)-presheaves on the full sub-(∞,1)-category of the $G$-orbit category on those orbits corresponding to compact subgroups (e.g. finite subgroups for $G$ discrete, details in DHLPS 19, Remark 3.1.12).

Related concepts

• equivariant stable homotopy theory

• global equivariant homotopy theory

References

The concept was introduced in

• Dieter Degrijse, Markus Hausmann, Wolfgang Lück, Irakli Patchkoria, Stefan Schwede, Proper equivariant stable homotopy theory, Memoirs of the AMS [arXiv:1908.00779]

following

• James Davis, Wolfgang Lück, Spaces over a Category and Assembly Maps in Isomorphism Conjectures in K- and L-Theory, K-Theory 15:201–252, 1998 (pdf)