# The spectrum of simplicial volume

## TL;DR

This paper explores the range of simplicial volumes in high-dimensional manifolds, showing density in all dimensions above 3 and rational realizability in dimension 4, using group homology and manifold construction techniques.

## Contribution

It introduces new methods to construct manifolds with prescribed simplicial volumes and establishes density results for all dimensions above 3, including rational values in dimension 4.

## Key findings

- Simplicial volumes are dense in $
_{\geq 0}$ for dimensions > 3.
- Every non-negative rational number is realizable as a simplicial volume in dimension 4.
- Group theoretic results connect stable commutator length to homology semi-norms.

## 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 $\mathbb{R}_{\geq 0}$. 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.