Citation
Das, Kaustuv Mukul (1994) Homotopy and homology of psubgroup complexes. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd06032004143153
Abstract
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In this thesis we analyzed the simple connectivity of the Quillen complex at [...] for the classical groups of Lie type. In light of the SolomonTits 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 CohenMacaulay 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 CohenIVlacaulay 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 nconnected implies X is nconnected.
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 pgroup 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 CohenMacaulay of dimension n2.
(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 ndimensional unitary space over Fq2 is simply connected.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Degree Grantor:  California Institute of Technology 
Division:  Physics, Mathematics and Astronomy 
Major Option:  Mathematics 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  12 April 1994 
Record Number:  CaltechETD:etd06032004143153 
Persistent URL:  http://resolver.caltech.edu/CaltechETD:etd06032004143153 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  2418 
Collection:  CaltechTHESIS 
Deposited By:  Imported from ETDdb 
Deposited On:  04 Jun 2004 
Last Modified:  26 Dec 2012 02:51 
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