Protocorks and monopole Floer homology

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 Szabó 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.

arXiv comment1) Strengthened Corollary 1.3. 2) The proof of Theorem 1.1.(b) is now independent from the results on moduli spaces of reducible monopoles of Section 5 so the proof of Theorem.1.1 depends only on formal properties of Floer homology. The results of Sections 5 are now presented in the introduction