# On intersection forms of definite 4-manifolds bounded by a rational homology 3-sphere

## TL;DR

This paper establishes finiteness results for the intersection forms of definite 4-manifolds bounded by a rational homology 3-sphere, using Donaldson's theorem and correction term invariants, with applications to spherical 3-manifolds.

## Contribution

It proves that only finitely many negative definite lattices can be realized as intersection forms of smooth 4-manifolds bounded by a given rational homology 3-sphere, extending previous constraints.

## Key findings

- Finiteness of negative definite lattices bounded by a rational homology 3-sphere.
- Application of Donaldson's diagonalization theorem and correction terms.
- All spherical 3-manifolds satisfy the finiteness property.

## Abstract

We show that, if a rational homology 3-sphere $Y$ bounds a positive definite smooth 4-manifold, then there are finitely many negative definite lattices, up to the stable-equivalence, which can be realized as the intersection form of a smooth 4-manifold bounded by $Y$. To this end, we make use of constraints on definite forms bounded by $Y$ induced from Donaldson's diagonalization theorem, and correction term invariants due to Fr\o yshov, and Ozsv\'ath and Szab\'o. In particular, we prove that all spherical 3-manifolds satisfy such finiteness property.