Subject: question on (co)homology of certain function spaces
Date: Thu, 11 Apr 2002 23:25:48 -0400 (EDT)
From: Claude Schochet
To: "DONALD M. DAVIS"
Here's one for toplist. Thanks.
Suppose that X is a compact space (separable metric, if you wish; finite
CW if you insist) and U_n denotes the n by n unitary matrices. Then I may
form the function space F(X,U_n) with the compact-open topology (no
basepoints.)
How does one go about computing the homology (cohomology) of this space?
Is the answer some well-known functor on X? Does the answer depend only
upon the homology (cohomology) of X? The space F(X,U_n) is a topological
group via point-wise operations and hence these results should give
information as Hopf algebras.
What is the homology (cohomology) of the classifying space of F(X, U_n)?
I would be interested both in specific examples and general results (e.g.,
appropriate spectral sequences that converge.) Results re what happens as
n goes to infinity are of special interest. (A moment's consideration of
the case X = point leads to Bott periodicity and Chern classes and
demonstrates nicely that the general situation is not trivial.)
Thanks!
Claude