Geometric obstructions for $\xi$-fillings of 3-manifolds

TL;DR

This paper develops a three-stage obstruction theory for realizing 4-manifolds with a specified boundary 3-manifold, introducing a new tertiary obstruction linked to Wall's quadratic self-intersection form.

Contribution

It introduces a novel tertiary obstruction in the realization problem for 4-manifolds, providing a geometric interpretation via Wall's quadratic self-intersection form.

Findings

Established a three-stage obstruction framework.

Identified a new tertiary obstruction for 4-manifold realization.

Connected the tertiary obstruction to Wall's quadratic form.

Abstract

We consider the realisation problem for normal 1-types of 4-manifolds with a given boundary. More precisely, given a normal 1-type $\xi$ and closed 3-dimensional $\xi$-manifold $Y$, does there exist a compact 4-dimensional $\xi$-manifold with boundary $Y$? We describe a three stage obstruction theory for the existence of such a 4-manifold, with our main contribution being a `tertiary' obstruction that we describe geometrically via Wall's quadratic self-intersection form.