On the simplicial volume and the Euler characteristic of (aspherical) manifolds

TL;DR

This paper investigates whether the vanishing of simplicial volume in aspherical manifolds implies the vanishing of Euler characteristic, exploring various versions, strategies, and related open problems in topology.

Contribution

It provides a comprehensive analysis of Gromov's question, introduces new variations, and examines the additivity and functorial properties of simplicial volume in different cobordism categories.

Findings

Simplicial volume acts as a symmetric monoidal functor on the amenable cobordism category.

The fundamental group of the 4-dimensional amenable cobordism category is not finitely generated.

Counterexamples exist among spaces homology equivalent to manifolds.

Abstract

A well-known question by Gromov asks whether the vanishing of the simplicial volume of oriented closed connected aspherical manifolds implies the vanishing of the Euler characteristic. We study various versions of Gromov's question and collect strategies towards affirmative answers and strategies towards negative answers to this problem. Moreover, we put Gromov's question into context with other open problems in low- and high-dimensional topology. A special emphasis is put on a comparative analysis of the additivity properties of the simplicial volume and the Euler characteristic for manifolds with boundary. We explain that the simplicial volume defines a symmetric monoidal functor (TQFT) on the amenable cobordism category, but not on the whole cobordism category. In addition, using known computations of simplicial volumes, we conclude that the fundamental group of the 4-dimensional amenable cobordism category is not finitely generated. We also consider new variations of Gromov's question. Specifically, we show that counterexamples exist among aspherical spaces that are only homology equivalent to oriented closed connected manifolds.