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The simplicial volume of contractible 3-manifolds (2105.09035v1)
Published 19 May 2021 in math.GT, math.AT, and math.DG
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.