Orbit spaces of subgroup complexes, Morse theory, and a new proof of a conjecture of Webb (Q2701840)

Let \({\mathcal R}_p(G)\) denote the simplicial complex of chains in the poset of \(p\)-subgroups in a finite group \(G\). There is an action of \(G\) on \({\mathcal R}_p(G)\) by simplicial maps, with the action on vertices given by conjugation of subgroups. The conjecture of Webb, recently proven by \textit{P.~Symonds} [Comment. Math. Helv. 73, No. 3, 400-405 (1998; Zbl 0911.20020)], asserts that the orbit space of \({\mathcal R}_p(G)\) under this action is contractible. In the paper under review, the author gives an alternate proof of this result, using a discrete Morse theory argument. This proof reveals a direct relationship between Webb's conjecture, for a given complex of subgroups, and analogues of the Sylow theorems (development and conjugacy) for the given family of subgroups. Thus the author is able to establish the contractibility of the spaces of orbits for an algebraic group acting on its complexes of torus subgroups, unipotent subgroups, and parabolic strict subgroups.NEWLINENEWLINENEWLINEIn each case, the orbit space has the structure of a \(\Delta\)-simplicial complex, a regular CW complex in which each cell is isometric with a Euclidean simplex. A distance function in the original poset determines a height function on the vertices, which serves as an analogue of a Morse function in the piecewise linear setting. The subgroup complexes associated with an algebraic group may be infinite; in this case the height function takes values in the ordinal numbers. Sylow-type theorems for the specified classes of subgroups imply the contractibility of certain subcomplexes determined by the height function. Roughly speaking, the conjugacy of maximal subgroups of the given type implies that these ``descending links'' are cones, and that there is a single minimal vertex. It follows by transfinite induction on the height that the orbit complex is contractible.