The simplicial volume of contractible 3-manifolds

2021/05/19 by Bargagnati, Giuseppe, Frigerio, Roberto
#53C23 #57K10 (primary) #57K30 #57M25 #57M35 #57N65 (secondary) #Algebraic Topology (math.AT) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2105.09035

Abstract

We show that the simplicial volume of a contractible 3-manifold not homeomorphic to ℝ³ 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≥ 4 there exists a contractible n-manifold with vanishing simplicial volume not homeomorphic to ℝⁿ. We also compute the spectrum of the simplicial volume of irreducible open 3-manifolds.

Related

• Cubical simplicial volume of 3-manifolds
• Transcendental simplicial volumes
• Minimal volume and simplicial norm of visibility n-manifolds and compact 3-manifolds
• The spectrum of simplicial volume of non-compact manifolds
• Simplicial volume of compact manifolds with amenable boundary