Title: Homotopy properties of the Quillen complex 
of the symmetric group 

Author: Dr. R. Ksontini, 
Institut de Math\'ematiques, 
Universit\'e de Lausanne, 
CH-1015 Lausanne, 
Suisse

Abstract:
Given a finite group $G$ and a prime $p$, the  
Quillen complex of $G$, denoted $\Delta A_p(G)$, 
is the order complexof the poset ($=$partially 
ordered set) $A_p(G)$ of nontrivial elementary 
abelian $p$-subgroups of $G$. This simplicial
complex with the action of $G$ by conjugation  
carries the information inherent in the $p$-local 
structure of $G$ and this is where lies its interest. 
If $G$ is the symmetric group $S_n$, the homotopy 
type of the poset $A_p(S_n)$, that is that of the 
simplicial complex $\Delta A_p(S_n)$, is  not known. 
The purpose of this work is to determine, through 
the study of subcomplexes denoted $\Delta D_p(n)$ 
and $\Delta T_p(n)$, already considered by Bouc, 
some homotopy properties of the Quillen complex 
$\Delta A_p(S_n)$.  Among other things, I 
determined for which values of $p$ and $n$ the 
poset $A_p(S_n)$ is simply connected. Moreover, 
if it is nontrivial, the fundamental group of the 
poset $A_p(S_n)$ is easy to calculate, except when 
$3p\leq n\leq 3p+1$ with $p\geq 3$. In this last 
case, the question remains open. 

Current status: thesis