Discrete Morse theory on $\Omega S^2$

TL;DR

This paper applies discrete Morse theory to a simplicial model of the loop space of S^2, identifying critical cells and computing homology, thus providing a combinatorial approach to classical homotopy results.

Contribution

It introduces a discrete Morse theoretic framework on Milnor's simplicial model of the loop space of S^2, linking combinatorial and algebraic topology.

Findings

Identified critical cells in the discrete Morse complex of the loop space

Computed the boundary operator on these cells

Recovered the first homology of the loop space of S^2

Abstract

A classical result in Morse theory is the determination of the homotopy type of the loop space of a manifold. In this paper, we study this result through the lens of discrete Morse theory. This requires a suitable simplicial model for the loop space. Here, we use Milnor's $\textrm{F}^+\textrm{K}$ construction to model the loop space of the sphere $S^2$, describe a discrete gradient on it, and identify a collection of critical cells. We also compute the action of the boundary operator in the Morse complex on these critical cells, showing that they are potential homology generators. A careful analysis allows us to recover the calculation of the first homology of $\Omega S^2$.