  We extend the Main Theorem of Aschbacher and Smith on Quillen's 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
nontrivial 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 $p=2$.
