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.

This work is based on the author's Master's thesis at Universität Augsburg. The author would like to thank his thesis advisor Bernhard Hanke for his invaluable help and support, and the referee for their helpful remarks.