Discrete Morse theory for manifolds with boundary (Q2844853)

Discrete Morse theory was originally developed by Robin Forman, and it has proven to be an extremely powerful tool in diverse areas of mathematics. In the paper under review, the author develops a version of discrete Morse theory for manifolds with boundary. The paper begins with an excellent introduction giving a survey of (classical smooth) Morse theory, analogous results in discrete Morse theory, and open questions. This sets up for the ``dual'' approach given in the paper. Let \(f\) be a discrete Morse function on a \(d\)-pseudo-manifold \(M\). Then \(f\) is said to be boundary critical if all the boundary faces of \(M\) are critical cells of \(f\). A manifold without boundary is said to be endo-collapsible if it admits a discrete Morse function with exactly \(2\) critical cells. A manifold with boundary is called endo-collapsible if it admits a discrete Morse function whose critical cells are all of the boundary cells plus a single interior cell.NEWLINENEWLINEThere are several important theorems that the author is able to show.NEWLINENEWLINETheorem: All endo-collapsible manifolds with boundary are balls.NEWLINENEWLINEThis is an analogous result to Whitehead's result that ``Every collapsible PL \(d\)-manifold is a ball'' but without the PL condition (it is unknown whether the PL condition is necessary in Whitehead's Theorem). The author provides a nice result about the barycentric subdivision of a constructible ball.NEWLINENEWLINETheorem: All constructible manifolds with boundary are endo-collapsible balls. Barycentric subdivision of simplicial constructible balls are collapsible.NEWLINENEWLINEAlso, endo-collapsibility of a pseudo-manifold \(M\) can be characterized in terms of the endo-collapsibility of the cone over \(M\). The author proves that the cone over \(M\) is endo-collapsible if and only if \(M\) is endo-collapsible.NEWLINENEWLINEChapter 2 gives a lengthy but helpful refresher on background results and terminology including polytopal complexes, PL manifolds, CW complexes, and discrete Morse theory.NEWLINENEWLINEChapter 3 begins by showing by showing that results analogous to classical discrete Morse theory hold in the boundary-critical case e.g. the relative Morse inequalities for a boundary-critical discrete Morse function. The author proves that endo-collapsible manifolds are either spheres or balls, obtaining in the process an alternative proof of Forman's theorem that if \(f\) is a discrete Morse function with \(2\) critical cells on a \(d\)-manifold \(M\) without boundary, then \(M\) is a \(d\)-sphere. There are also results in this chapter concerning the (good) behavior of boundary-critical discrete Morse functions under patching and barycentric subdivision.NEWLINENEWLINEChapter 4 studies applications of the author's theory to classical combinatorial topology; for example, to knotted \(3\)-manifolds of the kind studies by Goodrick and Bing.