Abstract
Let $\pi$ be a group equipped with an action of a second group $G$ by automorphisms. We define the equivariant cohomological dimension ${\sf cd}_G(\pi)$, the equivariant geometric dimension ${\sf gd}_G(\pi)$, and the equivariant LusternikSchnirelmann category ${\sf cat}_G(\pi)$ in terms of the Bredon dimensions and classifying space of the family of subgroups of the semidirect product $\pi\rtimes G$ consisting of subconjugates of $G$. When $G$ is finite, we extend theorems of EilenbergGanea and StallingsSwan to the equivariant setting, thereby showing that all three invariants coincide (except for the possibility of a $G$group $\pi$ with ${\sf cat}_G(\pi)={\sf cd}_G(\pi)=2$ and ${\sf gd}_G(\pi)=3$). A main ingredient is the purely algebraic result that the cohomological dimension of any finite group with respect to any family of proper subgroups is greater than one. This implies a StallingsSwan type result for families of subgroups which do not contain all finite subgroups.
Original language  English 

Journal  Groups, Geometry and Dynamics 
Volume  16 
Issue number  3 
DOIs  
Publication status  Published  29 Oct 2022 
Keywords
 55N91
 20J05 (Primary)
 55M30
 20E36 (Secondary)
 equivariant group cohomology
 equivariant Lusternik–Schnirelmann category
 classifying spaces
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Mark Grant
 School of Natural & Computing Sciences, Mathematical Science  Senior Lecturer
Person: Academic