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The title pretty much says it. In a follow-up to my question about alternating groups, does anyone know of a "geometric" model for $BGL_n(F_q)$? By "geometric" I mean "a space you would have heard about even if you aren't studying classifying spaces"; I mean to exclude general constructions such as standard nerve constructions, infinite joins, intractable quotients of frame bundles, etc.

edited Jul 31 2010 at 4:15

asked Jul 29 2010 at 19:30

Dev Sinha
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I have a couple thoughts about possible sources of such models. One is that sometimes models of BAut_C X where C is some category can be of the form "embeddings of X in a universal, contractible C-object." Examples include symmetric groups (BS_n = embeddings of points in R^\infty), general linear groups (linear embeddings of a vector space in a universal vector space (also works for G-actions)), and diffeomorphism groups (subsets of R^\infty diffeomorphic to a fixed manifold = embeddings modulo diffeomorphisms). Of course, universal F_q vector spaces are discrete... – Dev Sinha Jul 31 2010 at 4:22

The other thought is that there might be a poset/ simplicial construction which is geometrically defined - say built from the poset of finite-dimensional subspaces of (F_q)^\infty under inclusion or perhaps from some matroid theory. – Dev Sinha Jul 31 2010 at 5:20

Richard Borcherds · Answer 1 · 2010-07-31 14:18:00Z UTC

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Quillens' paper on the Adams conjecture (doi:10.1016/0040-9383(71)90018-8) almost gives an answer. He maps a limit of spaces BGL_n(F_q) to BU and shows that it is not far from an isomorphism. This is related to the plus construction, but cant remember the details offhand. The space BU in turn can be described in terms of Grassmannians.

answered Jul 31 2010 at 14:18

Richard Borcherds
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Thanks. As I recall, this result is stable (as you say), not valid at the prime p where q = p^r, and relies on some modular representation theory to construct the map. I am hoping (naively) that there might be a better model waiting to be found. – Dev Sinha Aug 2 2010 at 17:12

Geometric models for classifying spaces of $GLn(Fq)$.

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