Homology cobordism and classical knot invariants

Abstract

In this paper, we define and investigate \( \mathbb{Z}_2 \)-homology cobordism invariants of \( \mathbb{Z}_2 \)-homology 3-spheres which turn out to be related to classical invariants of knots. As an application, we show that many lens spaces have infinite order in the \( \mathbb{Z}_2 \)-homology cobordism group and we prove a lower bound for the slice genus of a knot on which integral surgery yields a given \( \mathbb{Z}_2 \)-homology sphere. We also give some new examples of 3-manifolds which cannot be obtained by integral surgery on a knot.