A triple torsion linking form and 3-manifolds in $S^4$

TL;DR

This paper introduces a new triple linking form for rational homology 3-spheres, relates it to higher order intersection forms, and applies it to obstruct embeddings in 4-spheres, extending classical results.

Contribution

It defines a novel triple linking form on homology, connects it with higher order intersection forms, and develops new embedding obstructions for rational homology spheres in $S^4$.

Findings

Defined a triple linking form for rational homology 3-spheres.

Connected the form to higher order intersection forms on bounding 4-manifolds.

Provided new obstructions for embedding 3-manifolds in 4-spheres.

Abstract

Given a rational homology 3-sphere $M$, we introduce a triple linking form on $H_1(M; \mathbb{Z})$, defined when the classical torsion linking pairing of three homology classes vanishes pairwise. If $M$ is the boundary of a simply-connected 4-manifold $N$, the triple linking form can be computed in terms of the higher order intersection form on $N$, introduced by Matsumoto. We use these methods to formulate an embedding obstruction for rational homology spheres in $S^4$, extending a 1938 theorem of Hantzsche.