Decomposing Perfect Discrete Morse Functions on Connected Sums of 3-manifolds

TL;DR

This paper demonstrates that perfect discrete Morse functions on connected sums of 3-manifolds can be decomposed into functions on the individual summands, with an explicit construction of the separating sphere.

Contribution

It provides a method to decompose perfect discrete Morse functions on connected sums of 3-manifolds into functions on each component, including an explicit construction of the separating sphere.

Findings

Decomposition of perfect discrete Morse functions on connected sums.

Explicit construction of the separating sphere.

Application to 3-manifold topology.

Abstract

In this paper, we show that if a closed, connected, oriented 3-manifold M = M1#M2 admits a perfect discrete Morse function, then one can decompose this function as perfect discrete Morse functions on M_1 and M_2. We also give an explicit construction of a separating sphere on M corresponding to such a decomposition.