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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 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
