Some results on Quillen’s conjecture via equivalent-poset techniques

We extend the main theorem of Aschbacher and Smith on the Quillen conjecture from p>5 to the remaining odd primes p = 3,5 . In the process, we develop further combinatorial and homotopical methods for studying the poset of non-trivial elementary abelian p -subgroups of a finite group. The techniques lead to a number of further results on the conjecture, often reducing dependence on the CFSG; in particular, we also provide some partial results toward the case of p=2 .