Equivariant collapses and a conjecture of Quillen

JONATHAN ARIEL BARMAK

Depto. de Matemática, FCEyN, UBA.

Conferencia; Algebraic Topology Conference in Buenos Aires; 2008

In 1978, D. Quillen studies homotopy properties of the simplicial complex K(Sp(G))associated to the poset Sp(G) of nontrivial p-subgroups of a finite group G. If G hasa nontrivial normal p-subgroup, K(Sp(G)) is contractible. Quillen conjectures theconverse, which is still an open problem. In 1984, R.E. Stong attacks this conjec-ture from the finite space point of view and proves that G has a nontrivial normalp-subgroup if and only if the finite space Sp(G) is contractible.In this talk I will recall the relationship between the homotopy and simple ho-motopy theory of finite topological spaces and polyhedra and Stong´s approach tothe equivariant homotopy theory of finite spaces. I will introduce the notion ofG-collapse of simplicial complexes, which is an equivariant version of Whitehead´scollapses, and I will show how this is related to an analogous concept for finitespaces. As a consequence of these ideas we will deduce that G has a nontrivialnormal p-subgroup if and only if K(Sp(G)) is G-collapsible.