Niemeier lattices, smooth 4-manifolds and instantons

TL;DR

This paper explores the relationship between Niemeier lattices and smooth 4-manifolds, demonstrating that the set of lattices associated with a fixed homology 3-sphere can vary beyond rank considerations.

Contribution

It shows that the set of lattices from smooth 4-manifolds bounded by a homology 3-sphere can depend on more than just lattice rank, providing explicit examples involving Niemeier lattices.

Findings

Different homology 3-spheres can have distinct sets of associated lattices.

The sets of lattices can include specific Niemeier lattices of rank 24.

The relationship between 4-manifolds and lattices is more nuanced than previously understood.

Abstract

We show that the set of even positive definite lattices that arise from smooth, simply-connected 4-manifolds bounded by a fixed homology 3-sphere can depend on more than the ranks of the lattices. We provide two homology 3-spheres with distinct sets of such lattices, each containing a distinct nonempty subset of the rank 24 Niemeier lattices.