A different approach is to consider the rational homotopy groups of UA or, to be brave / reckless, to consider the homotopy groups of UA itself. We report on progress in the calculation of these functors for the cases A = M_n(C(X)), A a unital continuous trace C*-algebra, and more generally for A an algebra of sections of a locally trivial bundle of C*-algebras. The resultant spectral sequence obtained generalizes work of H. Federer and J. Rosenberg. Finally, we discuss work in progress and a conjecture relating differentials and Samelson products in the unitary groups.These results are joint work and work in progress with J. Klein, G. Lupton, N.C. Phillips, S. Smith, and E. Dror-Farjoun.
However, various analytic properties of the theory remained unclear, such as its behaviour under direct or inverse limits, and little was known about when it coincides with other topological group cohomology theories such as that defined using continuous cochains. This talk will describe some recent progress which establishes such continuity in various cases, and also gives isomorphisms to some of those other cohomology theories, based on a simple procedure for showing that cocycles in Moore's theory can be `regularized' to exhibit more structure than assumed a priori.
I will try to make the talk mostly self-contained, but some
familiarity with basic homological algebra would be helpful to
listener. It is based on joint work with Calvin Moore.
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