Almost flat K-theory of classifying spaces

TL;DR

This paper explores the relationship between almost flat K-theory classes on classifying spaces and quasi-representations of fundamental groups, establishing continuity and correspondence properties for these mathematical structures.

Contribution

It provides a rigorous framework linking almost flat bundles, K-theory, and quasi-representations, extending understanding of their interplay in topological and algebraic contexts.

Findings

Established continuity properties for the correspondence.

Proved a bijective correspondence between K-theory classes and quasi-representations.

Extended the theory to classifying spaces with finite models.

Abstract

We give a rigorous account and prove continuity properties for the correspondence between almost flat bundles on a triangularizable compact connected space and the quasi-representations of its fundamental group. For a discrete countable group $\Gamma$ with finite classifying space $B\Gamma$, we study a correspondence between between almost flat K-theory classes on $B\Gamma$ and group homomorphism $K_0(C^*(\Gamma))\to \mathbb{Z}$ that are implemented by pairs of discrete asymptotic homomorphisms from $C^*(\Gamma)$ to matrix algebras.