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