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.

Figures (7)

• Figure 1: From left to right: a symmetric protocork plumbing graph with sphere-number $1$, an asymmetric protocork plumbing graph with sphere-number $3$, the reflection of the previous example.
• Figure 2: From the left: embedding of $\Gamma$, the result of removal of small balls centered at the vertices $\Gamma'$, introduction of 0-framed, oriented circles.
• Figure 3: Figure \ref{['figure:clasps:a']}) two oriented circles joined by an edge. Figure \ref{['figure:clasps:b']}) replacement of a positive edge in the spanning tree $T$ with a clasp. Figure \ref{['figure:clasps:c']}) replacement of a negative edge in $T$ with a clasp. Figure \ref{['figure:clasps:d']}) replacement of a positive edge not in $T$.
• Figure 4: A protocork plumbing graph (Figure \ref{['figure:variousP:a']}), its plumbing (Figure \ref{['figure:variousP:b']}), its protocork (Figure \ref{['figure:variousP:c']}) and the reflection of the protocork (Figure \ref{['figure:variousP:d']}).
• Figure 5: The picture shows how we obtain the Akbulut cork \ref{['fig:PA:e']} by adding two 2-handles to the protocork \ref{['fig:PA:a']}. The dashed circles represent the meridian to the $A$-sphere (red) and to the $B$-sphere (black). In \ref{['fig:PA:b']} we add the 2-handles (blue). To obtain \ref{['fig:PA:c']} we cancel the $0$-framed blue with the dotted circle, and similarly after sliding the red to the blue we cancel the $-1$-framed blue with the remaining dotted circle. To obtain \ref{['fig:PA:d']} we apply the Lemma shown in \ref{['figure:Lemma']}. \ref{['fig:PA:d']} is clearly isotopic to \ref{['fig:PA:e']}.

Theorems & Definitions (86)

• Theorem 1.1
• Corollary 1.2
• proof : Proof of Corollary \ref{['corollary:MorganSzaboImprovement']}
• Corollary 1.3
• proof : Proof of Corollary \ref{['intro:corollaryCorkDiff']}
• Theorem 1.4
• Theorem 1.5
• Remark 1
• Definition 2.1: Protocork plumbing graph
• Definition 2.2