Ribbon cobordisms as a partial order

TL;DR

This paper demonstrates that ribbon rational homology cobordism induces a partial order on aspherical 3-manifolds, extending the concept of partial orders from knot theory to 3-manifold topology.

Contribution

It establishes a new partial order structure on aspherical 3-manifolds based on ribbon cobordisms, supporting a conjecture by Daemi, Lidman, Vela-Vick, and Wong.

Findings

Ribbon rational homology cobordism defines a partial order on aspherical 3-manifolds.

The result extends the partial order concept from knots to 3-manifolds.

Supports the conjecture by Daemi et al. regarding this partial order.

Abstract

We show that the notion of ribbon rational homology cobordism yields a partial order on the set of aspherical $3$-manifolds, thus supporting a conjecture formulated by Daemi, Lidman, Vela-Vick and Wong. Our proof is built on Agol's recent proof of the fact that ribbon concordance yields a partial order on the set of knots in the 3-sphere.