Almost flat bundles and homological invariance of infinite K-area

TL;DR

This paper generalizes the concept of almost flat bundles to simplicial complexes and demonstrates that the property of infinite K-area is a homological invariant depending only on the fundamental class image, enhancing understanding of geometric invariants.

Contribution

It introduces a new notion of almost flat bundles over simplicial complexes and proves its invariance under certain maps, linking infinite K-area to fundamental class images.

Findings

Almost flat bundles can be extended to simplicial complexes.

Infinite K-area depends only on the fundamental class image.

The invariance holds up to a constant factor.

Abstract

We extend the notion of an almost flat bundle over a closed Riemannian manifold to bundles over simplicial complexes, and prove that up to a constant factor, this notion is invariant under pullback via maps which induce isomorphisms on fundamental groups. As an application, we show that the property of having infinite K-area only depends on the image of the fundamental class under the classifying map of the universal cover.