Odd manifolds of small integral simplicial volume

TL;DR

This paper proves that only odd-dimensional spheres have an integral simplicial volume of 1, and demonstrates that computing the integral simplicial volume of manifolds is generally not feasible.

Contribution

It establishes a unique characterization of odd spheres via integral simplicial volume and provides an elementary proof of the non-computability of this invariant.

Findings

Odd-dimensional spheres are the only manifolds with integral simplicial volume 1.

Integral simplicial volume is generally non-computable.

Provides an elementary proof of non-computability.

Abstract

Integral simplicial volume is a homotopy invariant of oriented closed connected manifolds, defined as the minimal weighted number of singular simplices needed to represent the fundamental class with integral coefficients. We show that odd-dimensional spheres are the only manifolds with integral simplicial volume equal to 1. Consequently, we obtain an elementary proof that, in general, the integral simplicial volume of (triangulated) manifolds is not computable.