Integer surgeries rational homology cobordant to lens spaces

TL;DR

This paper explores the limitations on positive integer surgeries on knots in $S^3$ that yield manifolds rational homology cobordant to lens spaces, using advanced invariants and lattice techniques.

Contribution

It introduces new bounds on surgeries producing rational homology cobordant manifolds, extending previous results on lens space surgeries.

Findings

At most two positive integer surgeries produce lens spaces from a non-trivial knot.

New constraints on surgeries yielding rational homology cobordant manifolds.

Application of changemaker lattices and Heegaard Floer invariants to surgery problems.

Abstract

The Cyclic Surgery Theorem and Moser's work on surgeries on torus knots imply that for any non-trivial knot in $S^3$, there are at most two integer surgeries that produce a lens space. This paper investigates how many positive integer surgeries on a given knot in $S^3$ can produce a manifold rational homology cobordant to a lens space. Tools include Greene and McCoy's work on changemaker lattices which come from Heegaard Floer $d$-invariants, and Aceto-Celoria-Park's work on rational cobordisms and integral homology which is based on Lisca's work on lens spaces.