# Is barycentric subdivision of a collapsible, regular CW complex collapsible (non-evasive)?

Let $X$ be a finite, regular CW complex, and let $X'$ be its barycentric subdivision (i.e. the order complex of the face poset of $X$). Assume $X$ is collapsible.

Is $X'$ collapsible? Is $X'$ non-evasive?

By Theorem 2.10 of Welker the answer to both questions is positive if $X$ is a simplicial complex. But what about more general, regular CW complexes? I believe at least the first question has a positive and well-known answer, but is it stated explicitly anywhere in the literature?

The first question is answered (modulo some details) by Forman in

R Forman, Morse theory for cell complexes. Advances in Mathematics, 134 pp 90 - 145, (1998).

Theorem 12.1 shows that a discrete Morse function $f$ on a polyhedron $X$ induces a discrete Morse function $\hat{f}$ on the subdivision $\hat{X}$ produced by bisecting a single $d$-cell $\sigma$ of dimension $d$ into two $d$-cells $\sigma_1$ and $\sigma_2$ which share a $d-1$-face $\tau$, and that (among other things) this $\hat{f}$ has precisely the same number of critical cells as $f$.

There are three things to check:

1. $X$ is collapsible if and only if it admits a discrete Morse function with precisely one critical cell (this is true).

2. Barycentric subdivision of $X$ may be achieved via a sequence of bisections so that each intermediate step produces a regular CW complex (I think this is true), and

3. The proof of this theorem does not require anything more from our polyhedron $X$ than the fact that it is a finite regular CW complex (this is true).