Asymptotically flat Fredholm bundles and assembly

TL;DR

This paper generalizes previous results on almost flat bundles to asymptotically flat Fredholm bundles, establishing an index theorem that links their indices with asymptotic indices, advancing the understanding of the Strong Novikov Conjecture.

Contribution

It introduces the concept of asymptotically flat Fredholm bundles and proves an index theorem relating their indices to asymptotic indices, extending prior work on flat bundles.

Findings

Generalization of Dadarlat's theorem to infinite-dimensional bundles with Fredholm operators

Introduction of asymptotically flat Fredholm bundles and their asymptotic Fredholm representations

Proving special cases of the Strong Novikov Conjecture using these bundles

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. We show how to generalize Dadarlat's theorem to the case of an infinite-dimensional bundle equipped with a continuous family of Fredholm operators on the fibers. Along the way, we show that special cases of the Strong Novikov Conjecture can be proven if there exist sufficiently many almost flat bundles with Fredholm operator. To this end, we introduce the concept of an asymptotically flat Fredholm bundle and its associated asymptotic Fredholm representation, and prove an index theorem which relates the index of the asymptotic Fredholm bundle with the so-called asymptotic index of the associated asymptotic Fredholm representation.