Heegaard Floer correction terms and Dedekind-Rademacher sums

TL;DR

This paper provides a closed-form formula for Heegaard Floer correction terms of lens spaces using Dedekind sums, revealing new relationships with classical invariants and establishing bounds on correction term vanishing.

Contribution

It introduces a formula linking correction terms to Dedekind-Rademacher sums and derives bounds on vanishing correction terms in lens spaces.

Findings

Casson-Walker invariant equals the average of correction terms

Obstruction for equality and opposite sign correction terms

Upper bound on vanishing correction terms

Abstract

We derive a closed formula for the Heegaard Floer correction terms of lens spaces in terms of the classical Dedekind sum and its generalization, the Dedekind-Rademacher sum. Our proof relies on a reciprocity formula for the correction terms established by Ozsvath and Szabo. A consequence of our result is that the Casson-Walker invariant of a lens space equals the average of its Heegaard-Floer correction terms. Additionally, we find an obstruction for the equality and equality with opposite sign, of two correction terms of the same lens space. Using this obstruction we are able to derive an optimal upper bound on the number of vanishing correction terms of lens spaces with square order second cohomology.