TITLE: Status of Quillen's conjecture on elementary abelian p-groups
QUESTION [15 upvotes]: These are questions on D. Quillen's 1978 paper Homotopy properties of the poset of nontrivial p-subgroups of a group.
Let $G$ be a finite group, $p$ a prime number, $\mathcal S(G)$ the poset of non-trivial $p$-subgroups of $G$, and $\mathcal A (G)$ the poset of non-trivial elementary Abelian $p$-subgroups of $G$, both ordered by inclusion.

Question One: Is $\mathcal S(G)$ homotopic or weakly homotopic to $\mathcal A (G)$? (Whichever is true is a theorem of Quillen.) If the latter, can someone give a specific example showing that the two posets are not homotopic?

Quillen also proved that for $G$ solvable, $\mathcal A (G)$ is contractible if and only if $G$ has a non-trivial normal $p$-subgroup.
Conjecture (Quillen): $\mathcal A (G)$ is contractible if and only if $G$ has a non-trivial normal $p$-subgroup.

Question Two: Is this still open (I know it was a few years ago)? What are lines of attack on this problem? Have attempts to prove it led to any a priori unrelated work?

REPLY [16 votes]: The fact that $\mathcal{A}(G)\simeq\mathcal{S}(G)$ is normally proved using Quillen's Theorem A as explained by Dan, but here is another argument that I like better.  (I don't think it is in the literature.) For any poset $X$, let $sX$ be the poset of nonempty chains in $X$, ordered by inclusion.  There is a homeomorphism $m:|sX|\to |X|$ by barycentric subdivision.  There is also a poset map $\max:sX\to X$, and $|\max|$ is homotopic to $m$ (by a linear homotopy) so it is a homotopy equivalence.  Next, if $P$ is a nontrivial $p$-group then the group $TP=\{g\in ZP: g^p=1\}$ is nontrivial (by a standard lemma) and elementary abelian.  Unfortunately, the map $T:\mathcal{S}(G)\to\mathcal{A}(G)$ is not order-preserving.  However, suppose we have a chain $C=(P_0\leq\dotsb\leq P_r)\in s\mathcal{S}(G)$.  When $i\leq j$ we note that $TP_i\leq P_j$ and $TP_j$ is central in $P_j$ so $TP_i$ commutes with $TP_j$.  It follows that the product $UC=TP_0.TP_1.\dotsb.TP_r$ is again elementary abelian.  This defines a poset map $U:s\mathcal{S}(G)\to\mathcal{A}(G)$ and thus a map $|U|:|\mathcal{S}(G)|\simeq|s\mathcal{S}(G)|\to|\mathcal{A}(G)|$.  If we let $i:\mathcal{A}(G)\to\mathcal{S}(G)$ denote the inclusion then $i\circ U\leq\max$ so $|i|\circ|U|$ is homotopic to $|\max|$ and thus to $m$.   We also have $U\circ si=\max:s\mathcal{A}(G)\to\mathcal{A}(G)$.  It follows that $|i|$ is a homotopy equivalence as claimed.