Towards the Homotopy Type of the Morse Complex

Mathematicians have long been interested in studying the properties of simplicial complexes. In 1998, Robin Forman developed gradient vector fields as a tool to study these complexes. Having gradient vector fields to study these simplicial complexes, in 2005, Chari and Joswig discovered the Morse complex, a complex consisting of all gradient vector fields on a fixed complex. Although the Morse complex has been studied since 2005, there is little information regarding its homotopy type for different simplicial complexes. Pursuing our curiosity of the topic, we extend a result by Ayala et. al., stating that the pure Morse complex of a tree is strongly collapsible. We also extend a result by Kozlov to show that a path with vertices equal to three multiplied by some number is strongly collapsible. Additionally, we provide alternate proofs for the results by Ayala et. al. as well as Kozlov. Furthermore, we realize cocktail-party graphs at the 1-skeleton of the core of the Morse complex of some paths, compute the homotopy type for centipede graphs, cycles with a single leaf, and some paths with a single leaf. By using multiple partitioning and matching strategies, we provide a framework to pursue homotopy types of more involved Morse complexes