Asymptotically Flat Fredholm Bundles and Assembly

Abstract

Almost flat finitely generated projective Hilbert C*-module bundles were successfully used by Hanke and Schick to prove special cases of the Strong Novikov Conjecture. Dadarlat later showed that it is possible to calculate the index of a K-homology class $\eta\in K_*(M)$ twisted with an almost flat bundle in terms of the image of $\eta$ under Lafforgue's assembly map and the almost representation associated to the bundle. Mishchenko used flat infinite-dimensional bundles equipped with a Fredholm operator in order to prove special cases of the Novikov higher signature conjecture.