2019/12/03 by Grant, Mark, Meir, Ehud, Patchkoria, Irakli
#20E36 (Secondary) #20J05 (Primary) #55M30 #55N91 #Algebraic Topology (math.AT) #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.1912.01692
Let π be a group equipped with an action of a second group G by automorphisms. We define the equivariant cohomological dimension \sf cdG(π), the equivariant geometric dimension \sf gdG(π), and the equivariant Lusternik-Schnirelmann category \sf catG(π) in terms of the Bredon dimensions and classifying space of the family of subgroups of the semi-direct product π\rtimes G consisting of sub-conjugates of G. When G is finite, we extend theorems of Eilenberg-Ganea and Stallings-Swan to the equivariant setting, thereby showing that all three invariants coincide (except for the possibility of a G-group π with \sf catG(π)=\sf cdG(π)=2 and \sf gdG(π)=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 Stallings-Swan type result for families of subgroups which do not contain all finite subgroups.