Lattices and correction terms

TL;DR

This paper links lattice embedding obstructions to topological properties of 3- and 4-manifolds using correction terms, providing a new perspective on Donaldson's theorem and Heegaard Floer invariants.

Contribution

It introduces lattice correction terms associated with metabolizing subgroups that determine lattice embeddings and relates these to topological obstructions for bounding rational homology balls.

Findings

Lattice embedding problems are characterized by correction terms for each metabolizing subgroup.

The obstruction for a rational homology 3-sphere to bound a rational homology 4-ball is rephrased via Donaldson's theorem.

Vanishing Heegaard Floer correction terms imply the Donaldson obstruction also vanishes under mild conditions.

Abstract

Let L be a nonunimodular definite lattice. Using a theorem of Elkies we show that whether L embeds in the standard definite lattice of the same rank is completely determined by a collection of lattice correction terms, one for each metabolizing subgroup of the discriminant group. As a topological application this gives a rephrasing of the obstruction for a rational homology 3-sphere to bound a rational homology 4-ball coming from Donaldson's theorem on definite intersection forms of 4-manifolds. Furthermore, from this perspective it is easy to see that if the obstruction to bounding a rational homology ball coming from Heegaard Floer correction terms vanishes, then (under some mild hypotheses) the obstruction from Donaldson's theorem vanishes too.