Birth and death in discrete Morse theory

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• 作者：Henry King^a ; ^{hck@math.umd.edu" class="auth_mail" title="E-mail the corresponding author} ; Kevin Knudson^b ; ^{kknudson@ufl.edu" class="auth_mail" title="E-mail the corresponding author} ; Neža Mramor Kosta^c ; ¹ ; ^{neza.mramor@fri.uni-lj.si" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 57Q99 ; 68U05 ; secondary ; 57R70 ; 58E05 ; 65D18 ; 65R99
• 刊名：Journal of Symbolic Computation
• 出版年：2017
• 出版时间：January-February 2017
• 年：2017
• 卷：78
• 期：Complete
• 页码：41-60
• 全文大小：1607 K

文摘

Suppose M is a finite cell decomposition of a space X   and that for 0=t₀<t₁<⋯<t_r=1$0=t_0<t_1<\cdots <t_r=1$ we have a discrete Morse function F_ti:M→R$F_{t_i}:M\to \mathbf{R}$. In this paper, we study the births and deaths of critical cells for the functions F_ti$F_{t_i}$ and present an algorithm for pairing the cells that occur in adjacent slices. We first study the case where the cell decomposition of X   is the same for each t_i$t_i$, and then generalize to the case where they may differ. This has potential applications in topological data analysis, where one has function values at a sample of points in some region in space at several different times or at different levels in an object.