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 mathbb {R}_{ge 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 l^1-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.
Highlights
The simplicial volume M of an orientable closed connected manifold M is a homotopy invariant that captures the complexity of representing fundamental classes by singular cycles with real coefficients
In Propostion 5.1, we show that the universal central extension E of T is a finitely presented group with H2(E; R) ∼= 0 and that every non-negative rational number may be realised by the stable commutator length of some element in E
The notion of simplicial volume of manifolds is based on the l1-semi-norm on singular homology
Summary
The simplicial volume M of an orientable closed connected (occ) manifold M is a homotopy invariant that captures the complexity of representing fundamental classes by singular cycles with real coefficients (see Sect. 2 for a precise definition and basic terminology). The simplicial volume M of an orientable closed connected (occ) manifold M is a homotopy invariant that captures the complexity of representing fundamental classes by singular cycles with real coefficients For large dimensions d, very little was known about the precise structure of the set SV(d) ⊂ R≥0 of simplicial volumes of occ d-manifolds. M ranges over all complete finite-volume hyperbolic 3-manifolds with toroidal boundary and where v > 0 is a constant (Example 2.5). The set of simplicial volumes of orientable closed connected d-manifolds is dense in R≥0.
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