Let $A$ be a $C^*$-algebra. Its unitary group, $UA$, contains a wealth of topological information about $A$. However, the homotopy type of $UA$ is unknown even for $A = M_2(\CC)$. There are various simplifications which have been considered. The first, well-traveled road, is to pass to $\pi_*(U(A\otimes \KK ))$ which is isomorphic (with a degree shift) to $K_*(A)$. This approach has led to spectacular success in many arenas, as is well-known.

A different approach is to consider $\pi _*(UA)\otimes\QQ $, the rational homotopy of $UA$ or, to be brave/reckless, to consider $\pi _*(UA) $ itself. We report on progress in the calculation of these functors for $A$ an algebra of sections of a locally trivial bundle of $C^*$-algebras over a compact metric space $X$ with $C^*$-algebra fibre $B$, so that $UA$ is the associated gauge group. If the bundle is trivial then $UA \cong F(X, UB)$ and the Federer spectral sequence (as generalized to compact metric spaces) may be used. Our interest is the case where the bundle is non-trivial, so that $A$ is a twisted algebra. We construct a spectral sequence converging to the homotopy of the gauge group $\pi _* (UA)$ with $E_2 \cong H^*(X; \pi _* (UB)) $ and a similar spectral sequence converging to $K_*(A)$.

In the case $X = S^k$ we produce a Wang sequence relating the homotopy of the gauge group $UA$ and of $UB$ and explain a conjecture identifying the differential in the gauge group sequence in terms of the classifying map of the bundle and a Samelson product. These results are joint work and work in progress with J. Klein, G. Lupton, N.C. Phillips, S. Smith, and E. Dror-Farjoun,