  This paper surveys topological results obtained from characteristic classes
built from the two types of traces on the algebra of pseudodifferential
operators of nonpositive order. The main results are the construction of a
universal $\hat A$-polynomial and Chern character that control the $S^1$-index
theorem for all circle actions on a fixed vector bundle over a manifold, and
$|\pi_1({\rm Diff}(M^5))| = \infty$, for ${\rm Diff}(M^5)$ the diffeomorphism
group of circle bundles $M^5$ with large first Chern class over projective
algebraic Kaehler surfaces.
