Multisections of 4-manifolds
Gabriel Islambouli, Patrick Naylor

TL;DR
This paper introduces multisections as a generalization of trisections for decomposing smooth, closed 4-manifolds into multiple pieces, enabling new operations and explicit descriptions of complex manifolds.
Contribution
It generalizes trisections to multisections, providing a framework for decomposing 4-manifolds into more than three parts and applying cut-and-paste techniques.
Findings
Multisections can describe any smooth, closed 4-manifold.
Implemented cork twists using multisections to relate exotic pairs.
Proved elliptic fibrations admit genus 3 multisections.
Abstract
We introduce multisections of smooth, closed 4-manifolds, which generalize trisections to decompositions with more than three pieces. This decomposition describes an arbitrary smooth, closed 4-manifold as a sequence of cut systems on a surface. We show how to carry out many smooth cut and paste operations in terms of these cut systems. In particular, we show how to implement a cork twist, whereby we show that an arbitrary exotic pair of smooth 4-manifolds admit 4-sections differing only by one cut system. By carrying out fiber sums and log transforms, we also show that the elliptic fibrations all admit genus 3 multisections, and draw explicit diagrams for these manifolds.
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Taxonomy
TopicsGeometric and Algebraic Topology · Advanced Combinatorial Mathematics · Algebraic Geometry and Number Theory
