Open aspherical manifolds not covered by the Euclidean space

TL;DR

This paper demonstrates that open aspherical manifolds in dimensions greater than three can have universal covers that are not homeomorphic to Euclidean space, challenging previous assumptions about their structure.

Contribution

It proves that such manifolds are tangentially homotopy equivalent to manifolds with non-Euclidean universal covers, revealing new topological properties.

Findings

Universal covers of certain open aspherical manifolds are not Euclidean spaces.

These manifolds are tangentially homotopy equivalent to manifolds with non-Euclidean universal covers.

The result applies to manifolds in dimensions greater than three.

Abstract

We show that any open aspherical manifold of dimension n>3 is tangentially homotopy equivalent to an n-manifold whose universal cover is not homeomorphic to the Euclidean space.