Homotopy equivalence of posets with a group action

Extensions of G-posets and Quillen's complex

“…The base group N = (Q XC P ) P is a normal subgroup of index p. This example is of interest because of counterexamples provided by Alperin to an incorrect conjecture made earlier by one of us [7,3.2] Now st/piCq^Cp) is the set of q Sylow p-subgroups of C q~A C p and so its homology is zero except in dimension zero, where it is V. As for si/ p (N), this is homotopy equivalent to the join srf p (C q XC P ) * • • • *£/ p (C q XC P ) with p factors by [4], and using the methods of [6] it is easy to see that Quillen's homotopy equivalence is equivariant for the action of G with C p permuting the factors. Since srf p (C q~A C p ) is just a set of q points, or in other words a wedge of (q -1) 0-spheres, the join of p copies of this space is a wedge of p(q -1) (p -1)-spheres, in which each C q X\C p factor of N permutes the corresponding set of (q -1) of these spheres.…”

Higher Limits via Subgroup Complexes

“…Taking C = S e p (G) and P = e in the above proof, we see that S p (G) and B p (G) are G-homotopy equivalent. This is a result of Bouc [15] and Thévenaz-Webb [73]. Remark 3.10.…”

“…[73] and Remark 4.3) that B e p (G) is equal to the opposite poset of the poset of parabolic subgroups in G. Hence the N P/P which occur in Theorem 1.3 will be of the form L J and, e.g., by Webb's Theorem 5.1, we may replace St * (L J ) with St L J in degree |J| − 1 (cf. Remark 2.4).…”

“…If P is not elementary abelian, then the inequalities H ≤ HΦ(P ) ≥ Φ(P ) show that |S p (G) <P | is N P -equivariantly contractible (see also [73]). (Here Φ(P ) denotes the Frattini subgroup of P ; alternatively, we use the inequalities To see that we can replace C by C ′ we use the same argument as in the proof of Theorem 1.1.…”

On the vertices of indecomposable summands of certain Lefschetz modules

“…The inclusion B.G/ Â S.G/ is a G-homotopy equivalence, see [11,Corollary,p. 50] and [50,Theorem 2]. Let D.G/ denote the subcollection of S.G/ consisting of the nontrivial p-centric and p-radical subgroups of G. This collection is not always homotopy equivalent to S.G/.…”