Brouwer degree, domination of manifolds, and groups presentable by products

TL;DR

This paper explores the concept of domination among manifolds via continuous maps of non-zero degree, introduces obstructions to domination, and investigates when manifolds can be dominated by products, linking these ideas to group properties.

Contribution

It connects the notion of manifold domination with group theoretic properties, specifically groups presentable by products, and provides examples of groups that are not presentable by products.

Findings

Obstructions to domination are characterized by Hopf and Gromov's simplicial volume.

Certain Coxeter groups are shown not to be presentable by products.

Abstract

For oriented connected closed manifolds of the same dimension, there is a transitive relation: $M$ dominates $N$, or $M \ge N$, if there exists a continuous map of non-zero degree from $M$ onto $N$. Section 1 is a reminder on the notion of degree (Brouwer, Hopf), Section 2 shows examples of domination and a first set of obstructions to domination due to Hopf, and Section 3 describes obstructions in terms of Gromov's simplicial volume. In Section 4 we address the particular question of when a given manifold can (or cannot) be dominated by a product. These considerations suggest a notion for groups (fundamental groups), due to D. Kotschick and C. L\"oh: a group is presentable by a product if it contains two infinite commuting subgroups which generate a subgroup of finite index. The last section shows a small sample of groups which are not presentable by products; examples include appropriate Coxeter groups.