The posets of p-subgroups of a finite group as finite topological spaces

KEVIN IVAN PITERMAN

Río de Janeiro

Encuentro; XXI Brazilian Topology Meeting; 2018

For a finite group G and a prime number p dividing its order, Sp(G) and Ap(G) denote respectively the poset of nontrivial p-subgroups of G and nontrivial elementary abelian psubgroups of G. They have been first studied by K. Brown [2] and D. Quillen [4] by means of the topological properties of their associated simplicial complexes K(Sp(G)) and K(Ap(G)). Quillen?s conjecture asserts that G has a nontrivial normal p-subgroup if and only if K(Sp(G)) is contractible. This conjecture has been widely studied in the last four decades by group theorists and algebraic topologists (See for example [1]).In [5] Stong studied the posets Sp(G) and Ap(G) as finite topological spaces (with an intrinsic topology). With this topology, they are not in general homotopy equivalent, although their associated simplicial complexes K(Sp(G)) and K(Ap(G)) always are. In this talk I will show that the finite space Ap(G) can be homotopically trivial but not contractible (this answers a question posted by Stong), and I will describe the contractibility of Sp(G) and Ap(G) as finite spaces in terms of algebraic properties of the group. These results recently appeared in [3]. I will also analyze a stronger formulation of a conjecture of P. Webb [6] in terms of finite spaces (the original conjecture was proved by P. Symonds), and prove particular cases of the strong conjecture using a combination of topological methods and fusion.[1] M. Aschbacher & S. D. Smith On Quillen?s conjecture for the p-groups complex. Ann. of Math. 137 (2), no. 3, 473-529, 1993.[2] K. Brown. Euler characteristics of groups: The p-fractional part. Invent. Math. 29, no. 1, 1-5, 1975.[3] E.G. Minian & K.I. Piterman. The homotopy types of the posets of p-subgroups of a finite group. Adv. Math. 328, 1217-1233, 2018.[4] D. Quillen. Homotopy properties of the poset of nontrivial p-subgroups of a group. Adv. Math. 28, 101-128, 1978.[5] R. E. Stong. Group actions on finite spaces. Discrete Math. 40, 95-100, 1984.[6] P. J. Webb. Subgroup complexes. The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986). Proc. Sympos. Pure Math., 47, Amer. Math. Soc., Providence, RI, 1987, 349-365.