On elliptic and quasiregularly elliptic manifolds

TL;DR

This paper explores the properties of elliptic and quasiregularly elliptic manifolds, providing new topological obstructions, examples of open manifolds with differing properties, and discussing their potential connection.

Contribution

It introduces new topological obstructions, constructs the first examples of open manifolds with differing elliptic properties, and investigates their possible equivalence.

Findings

Closed manifolds of both types have virtually abelian fundamental groups.

Constructed examples of open manifolds that are elliptic but not quasiregularly elliptic, and vice versa.

The connection and potential equivalence of the properties remain unresolved.

Abstract

In his book "Metric structures for Riemannian and non-Riemannian spaces", Gromov defined two properties of Riemannian manifolds, ellipticity and quasiregular ellipticity, and suggested that there may be a connection between the two. Since then, groups of researchers working independently have proved strikingly similar results about these two concepts. We obtain new topological obstructions to the two properties: most notably, we show that closed manifolds of both types must have virtually abelian fundamental group. We also give the first examples of open manifolds which are elliptic but not quasireguarly elliptic and vice versa. Whether there is a direct connection between these properties -- and, in particular, whether they are equivalent for closed manifolds -- remains elusive.