Fourier transforms and integer homology cobordism

TL;DR

This paper investigates the Fourier transform of Heegaard Floer d-invariants, revealing new properties and applications in 3-manifold and knot theory, including lens space cancellation and Alexander polynomial relations.

Contribution

It introduces a novel analysis of the Fourier transform of d-invariants and applies it to prove lens space cancellation and recover known knot polynomial results.

Findings

Lens spaces are cancellable in the 3-manifold cobordism monoid.

Revealed properties of the Fourier transform of d-invariants.

Recovered a theorem relating Alexander polynomials and reducible surgeries.

Abstract

We explore the Fourier transform of the Heegaard Floer $d$-invariants, which is particularly well-behaved with respect to connected sum. As corollaries, we show that lens spaces are cancellable in the monoid of 3-manifolds up to integer homology cobordism, and we recover a theorem of Gonz\'alez-Acu\~na--Short on Alexander polynomials of knots with reducible surgeries.