Discrete Morse theory on ΩS2

Abstract

One classical consequence of Morse theory is that it provides a description of a cell structure for a CW-complex having 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 ${\text{F}}^{+}\text{K}$ 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 $\Omega S^2$.