Monopoles and lens space surgeries

TL;DR

This paper uses monopole Floer homology to analyze Dehn surgeries on knots, proving certain manifolds cannot be obtained this way and providing insights into lens space surgeries and taut foliations.

Contribution

It introduces a surgery long exact sequence for monopole Floer homology and applies it to characterize knots with lens space surgeries and to identify manifolds lacking taut foliations.

Findings

Real projective three-space cannot result from non-trivial knot surgeries in S^3.

Monopole Floer homology detects the unknot.

Identifies manifolds that do not admit taut foliations.

Abstract

Monopole Floer homology is used to prove that real projective three-space cannot be obtained from Dehn surgery on a non-trivial knot in the three-sphere. To obtain this result, we use a surgery long exact sequence for monopole Floer homology, together with a non-vanishing theorem, which shows that monopole Floer homology detects the unknot. In addition, we apply these techniques to give information about knots which admit lens space surgeries, and to exhibit families of three-manifolds which do not admit taut foliations.