A characterisation of the Z^n + Z(\delta) lattice and definite nonunimodular intersection forms

TL;DR

This paper generalizes Elkies' theorem to nonunimodular lattices, providing a new criterion to determine whether certain rational homology 3-spheres can bound definite four-manifolds, with applications to surgeries on torus knots.

Contribution

It extends Elkies' theorem to nonunimodular lattices and develops a simple test for bounding definite four-manifolds in rational homology spheres.

Findings

Small positive surgeries on torus knots do not bound negative-definite four-manifolds.

The generalized theorem simplifies the detection of bounding four-manifolds.

New inequalities relate lattice properties to 3-manifold bounding conditions.

Abstract

We prove a generalisation of Elkies' theorem to nonunimodular definite forms (and lattices). Combined with inequalities of Froyshov and of Ozsvath and Szabo, this gives a simple test of whether a rational homology 3-sphere may bound a definite four-manifold. As an example we show that small positive surgeries on torus knots do not bound negative-definite four-manifolds.