A note on collapsibility of acyclic 2-complexes

TL;DR

This paper introduces a Morse-theoretic approach to determine when 2-dimensional acyclic simplicial complexes can be collapsed, using optimized combinatorial Morse functions to characterize collapsibility.

Contribution

It provides a novel Morse-theoretic characterization of collapsibility for acyclic 2-complexes based on normalized optimal Morse functions.

Findings

Morse-theoretic criteria for collapsibility

Use of normalized optimal combinatorial Morse functions

Characterization applicable to 2-dimensional acyclic complexes

Abstract

We present a Morse-theoretic characterization of collapsibility for 2-dimensional acyclic simplicial complexes by means of the values of normalized optimal combinatorial Morse functions.

Equations

$$h_f(\sigma )=\underset{g\in {\mathcal{Z}}_f}{\min }\{g(\sigma )\}.$$

$$h_f(\sigma )\le g(\sigma )<g(\tau )=h_f(\tau ).$$

$$g(\sigma )=h_f(\sigma )<h_f(\tau )\le g(\tau ).$$

$$h_f(\sigma )-1\ge h_f(\tau )>h_f(\eta )$$

$$g(\nu )=\{\begin{array}{ll}{h}_f(\nu )& \nu \ne \sigma \\ h_f(\sigma )-1& \nu =\sigma \end{array}$$

$$\mathfrak{N}(K,f):=\sum _{\sigma \in K}(-1{)}^{\dim (\sigma )}{h}_f(\sigma ).$$

$$0=\sum _{\begin{array}{c}\text{critical}\\ \text{vertices}\end{array}}{h}_f(v)-\sum _{\begin{array}{c}\text{critical}\\ \text{edges}\end{array}}{h}_f(e).$$

$$\mathfrak{N}(K):=\min \{|\mathfrak{N}(K,f)|:f:K\to \mathbb{R}\text{ optimal Morse function}\}.$$

$$0\to {\mathfrak{M}}_k\stackrel{{\partial }_k}{⟶}{\mathfrak{M}}_{k-1}\stackrel{{\partial }_{k-1}}{⟶}{\mathfrak{M}}_{k-2}\stackrel{{\partial }_{k-2}}{⟶}\cdots ,$$

$${\partial }_k(\tau )=\sum _{\sigma \in {\mathfrak{M}}_{k-1}}{\lambda }_{\sigma }^{\tau }\sigma ,$$

$$0=\sum _{j=1}^r{b}_j{\partial }_2({\sigma }_j),$$

$$\mathfrak{N}(L,f)=-\sum _{j=1}^k{h}_f(e_j)+\sum _{j=1}^k{h}_f({\sigma }_j)=\sum _{j=1}^k(h_f({\sigma }_j)-h_f(e_j))>0.$$

$$\tilde{\mathfrak{N}}(K):=\min \{\mathfrak{N}(L):L\text{ is a subdivision of }K\}.$$

Full text

E-mail address: ncapitelli@unlu.edu.ar

This research was partially supported by CONICET and the Department of Basic Sciences, UNLu (CDD-CB 148/18).

Let $K$ be a finite connected simplicial complex and let $f:K\rightarrow\mathbb{R}$ be a combinatorial Morse function over $K$. Let $\mathcal{Z}_{f}$ be the set of all combinatorial Morse functions $g:K\rightarrow\mathbb{Z}_{\geq 0}$ equivalent to $f$; i.e. inducing the same gradient field ($\mathcal{Z}_{f}\neq\emptyset$ by the finiteness of $K$). The normalization of $f$ is the map $h_{f}:K\rightarrow\mathbb{Z}_{\geq 0}$ defined by

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The function $h_{f}$ is also a combinatorial Morse function equivalent to $f$ (see Proposition I below). The purpose of this note is to give a characterization of collapsibility for $2$-dimensional acyclic simplicial complexes by means of the values of $h_{f}$. In what follows, we shall write $\sigma\prec\tau$ whenever $\sigma$ is an immediate face of $\tau$ (i.e. a proper face of maximal dimension).

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If now $h_{f}(\sigma)<h_{f}(\tau)$, let $g\in\mathcal{Z}_{f}$ be such that $h_{f}(\sigma)=g(\sigma)$. Then

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Since $f$ is equivalent to $g$ then $f(\sigma)<f(\tau)$. ∎

1. (1)

$h_{f}(\sigma)\geq\dim(\sigma)$* for all $\sigma\in K$.* 2. (2)

$h_{f}(\sigma)=0$* if and only if $\sigma$ is a critical vertex for $f$.* 3. (3)

If $\sigma\prec\tau$ and $f(\sigma)\geq f(\tau)$ then $h_{f}(\sigma)=h_{f}(\tau)$.

Item (2) follows from item (1) and the fact that lowering the value of any critical vertex in a function $g\in\mathcal{Z}_{f}$ produces again a combinatorial Morse function equivalent to $f$.

To see (3) suppose otherwise and let $\sigma$ be the simplex of minimal dimension satisfying $h_{f}(\sigma)>h_{f}(\tau)$. Note that $h_{f}(\tau)>h_{f}(\eta)$ for every $\eta\prec\sigma$. Indeed, if $\sigma^{\prime}$ is the other $\dim(\sigma)$-dimensional simplex containing $\eta$ as an immediate face then, by the choice of $\sigma$, we have $h_{f}(\eta)\leq h_{f}(\sigma^{\prime})<h_{f}(\tau)$. In particular

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for every $\eta\prec\sigma$. Therefore, the function

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is a combinatorial Morse function equivalent to $f$, thus contradicting the minimality of $h_{f}$. ∎

For a given combinatorial Morse function $f:K\rightarrow\mathbb{R}$ consider the number

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This definition is motivated by property (3) of Lemma II, which in turn implies that the sum may be taken over the critical simplices alone. We have the following result.

In the case of graphs, the other implication also holds.

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By Lemma II the first sum is zero and the second sum is positive if there is a critical edge. We conclude that $f$ has no critical edges. Since $G$ is connected there must be only one critical vertex. Hence $G$ is homotopy equivalent to CW with only a [math]-cell and thus it is a tree.∎

It is easy to see that Proposition IV does not hold in this generality for complexes of dimension greater than 1. Note however that the alleged functions appearing in these last two propositions can be taken to be optimal; i.e. they have the least possible number of critical simplices (among all combinatorial Morse functions over that complex). It is therefore natural to associate to a complex $K$ the number

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With this definition, Proposition III may be restated as follows: “If $K$ is collapsible then $\mathfrak{N}(K)=0$”. The converse of this statement does not hold in dimension greater than $1$ either (see Figure 1). However, the number $\mathfrak{N}$ can be used to characterize collapsibility for acyclic $2$-complexes. The main result of this note is the following.

Before we prove Theorem V recall that, given a combinatorial Morse function $f:K\rightarrow\mathbb{R}$, the Morse complex associated to $f$ is the chain complex of $\mathbb{R}$-vector spaces

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where $\mathfrak{M}_{k}$ is the span of the critical $k$-simplices of $f$. By [2, Theorem 8.2], this complex has the same homology with real coefficients as $K$. Also, [2, Theorem 8.10] shows that the boundary map $\partial_{k}:\mathfrak{M}_{k}\rightarrow\mathfrak{M}_{k-1}$ can be written

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where the coefficients $\lambda^{\tau}_{\sigma}$ depend on the set $\Gamma(\tilde{\sigma},\sigma)$ of gradient paths between $\sigma$ and the immediate faces $\tilde{\sigma}$ of $\tau$ (see [2, §8]). In particular, if $\Gamma(\tilde{\sigma},\sigma)=\emptyset$ for every $\tilde{\sigma}\prec\tau$ then $\lambda^{\tau}_{\sigma}=0$.

We also shall make use of the following classical result from Graph Theory (see e.g. [4]):

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for some $b_{i}\in\mathbb{R}$, not all zero. But in this case, $\sum_{j=1}^{r}b_{j}\sigma_{j}$ is a generating cycle of $H_{2}(\mathfrak{M}_{\ast},\partial_{\ast})\simeq H_{2}(L)$ and we reach a contradiction to our hypotheses. This proves that there exists a complete matching $\mathcal{M}$ in $G$. Order $A=\{e_{1},\ldots,e_{k}\}$ and $B=\{\sigma_{1},\ldots,\sigma_{k}\}$ so that $(e_{i},\sigma_{i})\in\mathcal{M}$ for every $i=1,\ldots,k$. By construction, there is a gradient path from a boundary edge of $\sigma_{i}$ to $e_{i}$ for every $i=1,\ldots,k$. In particular, $h_{f}(e_{i})<h_{f}(\sigma_{i})$ for every $i=1,\ldots,k$. We conclude that

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∎

It is straightforward to produce similar results for $PL$-collapsibility. A complex is $PL$-collapsible if it has a collapsible subdivision. For a complex $K$ one can define the number

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As a direct corollary to Theorem V we have the following result.

Acknowledgements

I would like to thank Gabriel Minian for many useful comments and suggestions.