Almost flat relative vector bundles and the almost monodromy correspondence

TL;DR

This paper introduces the concept of almost flatness for relative bundles on topological spaces, establishes their equivalence in topological and smooth contexts, and develops an almost monodromy correspondence linking these bundles to relative quasi-representations.

Contribution

It defines almost flatness for relative bundles, proves their equivalence in different senses, and establishes a new correspondence with relative quasi-representations.

Findings

Almost flatness in topological and smooth contexts are equivalent.

Constructs almost flat relative bundles using manifold enlargeability.

Establishes the almost monodromy correspondence between bundles and quasi-representations.

Abstract

In this paper we introduce the notion of almost flatness for (stably) relative bundles on a pair of topological spaces and investigate basic properties of it. First, we show that almost flatness of topological and smooth sense are equivalent. This provides a construction of an almost flat stably relative bundle by using the enlargeability of manifolds. Second, we show the almost monodromy correspondence, that is, a correspondence between almost flat (stably) relative bundles and (stably) relative quasi-representations of the fundamental group.