Linear independence in the rational homology cobordism group

TL;DR

This paper establishes simple homological criteria to determine when rational homology 3-spheres have infinite order or are linearly independent in the rational homology cobordism group, with implications for knot concordance.

Contribution

It introduces homological conditions for infinite order and linear independence in the cobordism group, connecting these to correction terms and knot theory.

Findings

Criteria for infinite order in the cobordism group

Conditions for linear independence of rational homology spheres

Applications to knot concordance via branched double covers

Abstract

We give simple homological conditions for a rational homology 3-sphere Y to have infinite order in the rational homology cobordism group, and for a collection of rational homology spheres to be linearly independent. These translate immediately to statements about knot concordance when Y is the branched double cover of a knot, recovering some results of Livingston and Naik. The statements depend only on the homology groups of the 3-manifolds, but are proven through an analysis of correction terms and their behavior under connected sums.