On $h$-cobordisms of complexity $2$

TL;DR

This paper investigates 5-dimensional h-cobordisms with complexity 2, computing monopole Floer homology and revealing obstructions to minimal complexity between exotic 4-manifolds, including new examples of high-complexity cobordisms.

Contribution

It introduces the first examples of high-complexity h-cobordisms between exotic 4-manifolds and analyzes their Floer homology and involution actions.

Findings

Computed monopole Floer homology for these cobordisms

Identified obstructions to minimal complexity in h-cobordisms

Constructed examples with arbitrarily large complexity

Abstract

We study $5$-dimensional $h$-cobordisms of Morgan-Szab\'o complexity $2$. We compute the monopole Floer homology and the action of the twisting involution of the protocork boundary associated with such $h$-cobordisms, obtaining an obstruction for $h$-cobordisms between exotic pairs to have minimal complexity. We construct the first examples of $h$-cobordisms of non-minimal, in fact, arbitrarily large, complexity between an exotic pair of closed, $1$-connected $4$-manifolds. Further applications include strong corks.