Saeed Tepsi: A Homotopy Theoretic Approach to Quillen’s Conjecture

Abstract.

In 1978 the American mathematician Daniel Quillen was studying the topological properties of a certain poset associated to a group. He provided necessary and sucient conditions for a solvable group to contain a normal \(p\)-subgroup and conjectured that it will also hold for any finite group. In this paper we will develop some of the techniques one will need to understand, restate and perhaps attack the conjecture. We will start by developing some of the theory of finite topological spaces, simplicial complexes and study their homotopy types. Then we turn our attention to groups, in particular we will look at the equivariant properties of the poset \(\operatorname{Sp}(G)\). We finish with a theorem that allows one to attack Quillen’s conjecture from a lot of different angles and a neat result by Brown on the Euler characteristic of \(\operatorname{Sp}(G)\).