{"title":"Discrete Morse theory on $ΩS^2$","authors":"Lacey Johnson, Kevin Knudson","doi":"arxiv-2407.12156","DOIUrl":null,"url":null,"abstract":"A classical result in Morse theory is the determination of the homotopy type\nof the loop space of a manifold. In this paper, we study this result through\nthe lens of discrete Morse theory. This requires a suitable simplicial model\nfor the loop space. Here, we use Milnor's $\\textrm{F}^+\\textrm{K}$ construction\nto model the loop space of the sphere $S^2$, describe a discrete gradient on\nit, and identify a collection of critical cells. We also compute the action of\nthe boundary operator in the Morse complex on these critical cells, showing\nthat they are potential homology generators. A careful analysis allows us to\nrecover the calculation of the first homology of $\\Omega S^2$.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"15 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Algebraic Topology","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2407.12156","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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 $\Omega S^2$.