Discrete Morse Theory for the Barycentric Subdivision

Abstract

We work with discrete Morse theory. Let F be a discrete Morse function on a simplicial complex L. We construct a discrete Morse function ∆(F ) on the barycentric subdivision ∆(L). The constructed function ∆(F ) "behaves the same way" as F , i. e. has the same number of critical simplexes and the same gradient path structure.

Discrete Morse Theory by Nicholas Scoville

“…topology of K since we could deform it to the edge e. Now, if we chose the 2-simplex τ in sd(σ) to be critical and employ the procedure in [22] without alteration, we would have the barycenter b paired with the edge f , yielding no critical cells in W in the interior of σ (since both b and f lie in S K ). In principle, this is not an issue since W will determine the homotopy type of |S K | ≃ |K|, but we will have lost track of the original critical cell σ from V K .…”

“…Consider the barycentric subdivision sd(X). In [22], A. Zhukova describes a procedure to induce a discrete gradient on sd(X) from a given gradient on X in such a way that there is a one-to-one correspondence between the critical cells of the two gradients. In fact, each critical cell in the subdivision will lie in the interior of a critical cell (of the same dimension) of the original gradient.…”

“…We will make a slight modification to the procedure in [22] to keep better track of the critical simplices in K. Suppose σ is a critical k-cell in V K and suppose the height of σ is i. We can choose any of the k-cells in sd(σ) to be critical, but in [22,Sec. 3.2] the author chooses a particular one based on a fixed ordering of the vertices of σ.…”

“…If β has no codimension-1 face that is the tail of a matched pair in K (aside from α, of course), perform the construction of [22,Sec. 3.1] to subdivide β and pair the simplices in sd(β) together.…”

“…for some vertex u. Perform the construction of [22], and then make the following modification. Zhukova's construction pairs the barycenter b with the edge bv k+1 , and it pairs the edge bz, where z is the barycenter of α ′ with the 2-simplex bzv k+1 .…”