Protocorks and monopole Floer homology

TL;DR

This paper introduces protocorks, a new class of 4-manifolds with boundary, and explores their role in relating exotic smooth structures, analyzing Seiberg-Witten invariants, and Floer homology, advancing understanding of 4-manifold topology.

Contribution

It defines protocorks and proves their significance in connecting exotic 4-manifolds, along with new theorems on Seiberg-Witten invariants and Floer homology related to protocork twists.

Findings

Protocorks can relate any exotic pair of simply connected closed 4-manifolds.

Theorems on the behavior of Seiberg-Witten invariants under protocork twists.

Splitting theorem for Floer homology of protocork boundaries.

Abstract

We introduce and study a class of compact 4-manifolds with boundary that we call protocorks. Any exotic pair of simply connected closed 4-manifolds is related by a protocork twist, moreover, any cork is supported by a protocork. We prove a theorem on the relative Seiberg-Witten invariants of a protocork before and after twisting and a splitting theorem on the Floer homology of protocork boundaries. As a corollary we improve a theorem by Morgan and Szab\'{o} regarding the variation of Seiberg-Witten invariants with an upper bound which depends only on the topology of the data. Moreover, we generalize the result that only the reduced Floer homology of a cork boundary contributes to the variation of the Seiberg-Witten invariants under a cork twist to more general cut and paste operations where the pieces involved are $1$-connected and homeomorphic relative to the boundary.