Multisections of 4-manifolds

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 $E(n)_{p,q}$ all admit genus 3 multisections, and draw explicit diagrams for these manifolds.