Handle decomposition complexity and representation spaces

TL;DR

This paper investigates the handle decomposition complexity of certain homology three-spheres that bound definite four-manifolds, utilizing the homology cobordism invariant  from instanton Floer homology to establish new constraints.

Contribution

It proves that some homology three-spheres bounding definite four-manifolds require complex handle decompositions, revealing new topological obstructions.

Findings

Certain homology three-spheres must have complex handle decompositions

The invariant  provides obstructions to simple bounding four-manifolds

Handle complexity relates to the homology cobordism invariant 

Abstract

We prove that there are homology three-spheres that bound definite four-manifolds, but any such bounding four-manifold must be built out of many handles. The argument uses the homology cobordism invariant $\Gamma$ from instanton Floer homology.