Reconstructing 4-manifolds from Morse 2-functions

TL;DR

This paper establishes minimal conditions under which 4-manifolds and their Morse 2-functions can be reconstructed from combinatorial diagrams, with implications for understanding manifold topology.

Contribution

It introduces minimal criteria for reconstructing 4-manifolds from Morse 2-functions using combinatorial diagrams, extending to other dimensions.

Findings

Reconstruction of 4-manifolds from combinatorial data

Minimal conditions on fold curves and fibers

Extension of methods to other dimensions

Abstract

Given a Morse 2-function $f: X^4 \to S^2$, we give minimal conditions on the fold curves and fibers so that $X^4$ and $f$ can be reconstructed from a certain combinatorial diagram attached to $S^2$. Additional remarks are made in other dimensions.