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Type of Document	Dissertation
Author	Das, Kaustuv Mukul
URN	etd-06032004-143153
Persistent URL	http://resolver.caltech.edu/CaltechETD:etd-06032004-143153
Title	Homotopy and homology of p-subgroup complexes
Degree	PhD
Option	Mathematics
Advisory Committee	Advisor Name Title Prof. Michael Aschbacher Committee Chair David G. Wales Committee Member Richard M. Wilson Committee Member Robert M. Guralnick Committee Member Shahriar Shahriari Committee Member
Keywords	none
Date of Defense	1994-04-12
Availability	unrestricted
Abstract NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. In this thesis we analyzed the simple connectivity of the Quillen complex at [...] for the classical groups of Lie type. In light of the Solomon-Tits theorem, we focused on the case where [...] is not the characteristic prime. Given (p,q) = 1. let dp(q) be the order of [...] in [...]. In this thesis we proved the following result: Main Theorem. When (p,q) = 1 we have the following results about the simple connectivity of the Quillen complex at p, Ap(G), for the classical groups of Lie type: 1. If G = GLn(q), dp(q) > 2 and mp(G) > 2, then Ap[...](G) is simply connected. 2. If G = [...], then: (a) Ap(G) is Cohen-Macaulay of dimension n - 1 if dp(q) = 1. (b) If nip(G) > 2 and dp(q) is odd, then Ap(G) is simply connected. 3. If G = [...], then: (a) Ap(G) is Cohen-IVlacaulay of dimension n - 1 if [...] and dp(q) = 1. (b) If mp(G) > 2 and dp(q) is odd, then ,Ap(G) is simply connected. (c) If n [...] 3, q [...] 5 is odd, and dp(q) = 2, then Ap(G)(> Z) is simply connected, where Z is the central subgroup of G of order p. In the course of analyzing the [...]-subgroup complexes we developed new tools for studying relations between various simplicial complexes and generated results about the join of complexes and the [...]-subgroup complexes of products of groups. For example we proved: Theorem A. Let [...] be a map of posets satisfying: (1) [...] is strict; that is,[...] (2)[...] (3) [...]connected for all [...] with [...]. Then Y n-connected implies X is n-connected. Theorem A provides us with a tool for studying [...] in terms of [...]. For example, we used this method to prove: Theorem 8.6. Let G = [...] where [...] is solvable and S is a p-group of symplectic type. Then [...]spherical. In this thesis we also generated a library of results about geometric complexes which do not arise as [...]-subgroup complexes. This library includes, but is not restricted to, the following: (l.) the poset of proper nondegenerate subspaces of a 2[...]-dimensional symplectic space -ordered by inclusion - is Cohen-Macaulay of dimension n-2. (2) If q is an odd prime power anal n [...] (with n [...] 5 if q = 3), then the poset of proper nondegenerate subspaces of an n-dimensional unitary space over Fq2 is simply connected.
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