The simplicial volume of contractible 3-manifolds

TL;DR

This paper proves that the simplicial volume of contractible 3-manifolds is infinite unless they are Euclidean space, and explores properties of higher-dimensional contractible manifolds and their simplicial volumes.

Contribution

It establishes the characterization of Euclidean space among contractible 3-manifolds via simplicial volume and investigates the existence of contractible manifolds with zero simplicial volume in higher dimensions.

Findings

Simplicial volume of non-ℝ³ contractible 3-manifolds is infinite.

Euclidean space uniquely has zero simplicial volume among contractible 3-manifolds.

In dimensions ≥4, there exist contractible manifolds with zero simplicial volume not homeomorphic to ℝⁿ.

Abstract

We show that the simplicial volume of a contractible 3-manifold not homeomorphic to $\mathbb{R}^3$ is infinite. As a consequence, the Euclidean space may be characterized as the unique contractible $3$-manifold with vanishing minimal volume, or as the unique contractible $3$-manifold supporting a complete finite-volume Riemannian metric with Ricci curvature uniformly bounded from below. On the contrary, we show that in every dimension $n\geq 4$ there exists a contractible $n$-manifold with vanishing simplicial volume not homeomorphic to $\mathbb{R}^n$. We also compute the spectrum of the simplicial volume of irreducible open 3-manifolds.