The spectrum of simplicial volume of non-compact manifolds

TL;DR

This paper investigates the range of simplicial volumes for non-compact manifolds, demonstrating that in dimensions four and higher, these volumes can be any non-negative real number or infinity, with specific considerations for tame and low-dimensional cases.

Contribution

It establishes that the set of simplicial volumes for open manifolds in dimension four and above is the entire non-negative extended real line, expanding understanding of their geometric invariants.

Findings

Simplicial volumes of open manifolds in dimension ≥4 cover all non-negative real numbers and infinity.

Analysis of tame open manifolds and low-dimensional examples.

The set of simplicial volumes is dense in [0, ∞].

Abstract

We show that, in dimension at least $4$, the set of locally finite simplicial volumes of oriented connected open manifolds is $[0, \infty]$. Moreover, we consider the case of tame open manifolds and some low-dimensional examples.