#### Submission Date

4-20-2023

#### Document Type

Paper

#### Department

Mathematics

#### Adviser

Nicholas Scoville

#### Committee Member

Christopher Sadowski

#### Committee Member

Christopher Tralie

#### Department Chair

Nicholas Scoville

#### External Reviewer

Joseph Grenier

#### Distinguished Honors

This paper has met the requirements for Distinguished Honors.

#### Project Description

In this thesis, we study possible homotopy types of four families of simplicial complexes–the Morse complex, the generalized Morse complex, the matching complex, and the independence complex–using discrete Morse theory. Given a simplicial complex, K, we can construct its Morse complex from all possible discrete gradient vector fields on K. A similar construction will allow us to build the generalized Morse complex while considering edges and vertices will allow us to construct the matching complex and independence complex. In Chapter 3, we use the Cluster Lemma and the notion of star clusters to apply matchings to families of Morse, generalized Morse, and matching complexes, computing their homotopy types. Notably, we show that the Morse complex of a subset of extended star graphs is homotopy equivalent to a wedge of spheres and the matching complex of a Dutch windmill graph is homotopy equivalent to a point, sphere, or wedge of spheres. In Chapter 4, we use a degenerate Hasse diagram and strong collapses to compute the homotopy type of many families of Morse complexes. Recognizably, we provide computations showing wedged complexes as suspensions and provide a sufficient condition for strongly collapsible Morse complexes. Lastly, in Chapter 5 we study chord diagrams–a largely unexplored topic–and provide insight into the possible homotopy types of the independence complex of intersection graphs of chord diagrams. We realize spheres and wedges of spheres as possible homotopy types and begin to explore what families of intersection graphs can be represented as a chord diagram. From here, most interesting, we show that ladder graphs can be represented as chord diagrams, and the independence complex of a ladder graph has the homotopy type of a sphere.

#### Recommended Citation

Donovan, Connor, "A Discrete Morse Approach for Computing Homotopy Types: An Exploration of the Morse, Generalized Morse, Matching, and Independence Complexes" (2023). *Mathematics Honors Papers*. 4.

https://digitalcommons.ursinus.edu/math_hon/4