The spectrum of simplicial volume

2019/04/09 by Heuer, Nicolaus, Loeh, Clara
#20J05 #57M07 #57N65 #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1904.04539

Abstract

New constructions in group homology allow us to manufacture high-dimensional manifolds with controlled simplicial volume. We prove that for every dimension bigger than 3 the set of simplicial volumes of orientable closed connected manifolds is dense in ℝ_{≥ 0}. In dimension 4 we prove that every non-negative rational number is the simplicial volume of some orientable closed connected 4-manifold. Our group theoretic results relate stable commutator length to the l¹-semi-norm of certain singular homology classes in degree 2. The output of these results is translated into manifold constructions using cross-products and Thom realisation.

Related

• The spectrum of simplicial volume of non-compact manifolds
• The spectrum of simplicial volume with fixed fundamental group
• Simplicial volume and essentiality of manifolds fibered over spheres
• The Proportionality Principle of Simplicial Volume
• Transcendental simplicial volumes