The simplicial volume of contractible 3-manifolds

Bargagnati, Giuseppe; Frigerio, Roberto

doi:10.1090/tran/8605

The simplicial volume of contractible 3-manifolds
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by Giuseppe Bargagnati and Roberto Frigerio PDF

Trans. Amer. Math. Soc. 375 (2022), 3305-3323 Request permission

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. In contrast, 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.

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