Lecture 8a. A primer on simplicial complexes and collapsibility

Before we can apply more advanced fixed point theorems to the Evasiveness Conjecture, we need a little background on simplicial complexes, and everything starts with simplices.

Simplices

It’s most intuitive to start with the geometric viewpoint, in which case an $n$ -simplex is defined to be the convex hull of $n+1$ affinely independent points in $\mathbb R^n$ . These points are called the vertices of the simplex. Here are examples for $n=0,1,2,3.$

simplices2

A simplicial complex is then a collection of simplices glued together along lower-dimensional simplices. More formally, if $S \subseteq \mathbb R^n$ is a (geometric) simplex, then a face of $S$ is a subset $F \subseteq S$ formed by taking the convex hull of a subset of the vertices of $S$ .

Finally, a (geometric) simplicial complex is a collection $\mathcal K$ of simplices such that

If $S \in \mathcal K$ and $F$ is a face of $S$ , then $F \in \mathcal K$ , and
If $S,S' \in \mathcal K$ and $S \cap S' \neq \emptyset$ , then $S \cap S'$ is a face of both $S$ and $S'$ .

Property (1) gives us downward closure, and property (2) specifies how simplices can be glued together (only along faces). For instance, the first picture depicts a simplicial complex. The second does not.

scexample
scnonexample1

There is also an abstract way to define a simplicial complex. An abstract simplicial complex is a ground set $X$ of vertices, together with a collection $\mathcal K$ of subsets of $X$ satisfying the axioms

$\mathcal K \neq \emptyset$
If $A \in \mathcal K$ and $A' \subseteq A$ , then $A' \in \mathcal K$ .

For instance, a standard (undirected) graph can be thought of as a 1-dimensional abstract simplicial complex. We can always realize an abstract complex as a geometric complex by taking $|X|$ affinely independent points in $\mathbb R^{|X|-1}$ and adding in the appropriate convex hulls. (Exercise: Every 1-dimensional complex can be realized in $\mathbb R^3$ , but not $\mathbb R^2$ .) In general, we will switch freely between the two notions. (Here is an interesting paper of Matousek, Tancer, and Wagner on the computational complexity of realizability.)

Contractibility and collapsibility

Say that a set $T \subseteq \mathbb R^n$ is contractible if there exists a continuous mapping $\Phi : T \times \lbrack 0,1\rbrack \to T$ such that

$\Phi(\cdot,0)$ is the identity on $T$ .
$\Phi(T,1) = \{p_0\}$ for some fixed $p_0 \in T$ .

In other words, $T$ can be contracted to a point in place. For instance, the closed unit ball in $\mathbb R^3$ is contractible while the unit sphere is not (think about it!) As another example, consider that a geometric realization of a graph (a 1-dimensional simplicial complex) is contractible if and only if the graph is a connected tree.

We now define two basic operations on simplicial complexes. The first involves removing a vertex

$\mathcal K \setminus v = \{ S \in \mathcal K : v \notin S \}.$

The second, called the link of $v$ in $\mathcal K$ is

$K / v = \{ S \in \mathcal K : v \notin S, S \cup \{v\} \in \mathcal K \}.$

For example, consider the following complex $\mathcal K$ , in which a distinguished simplex $A$ is marked.

complex Then we form the two simplices $\mathcal K \setminus x$ and $\mathcal K / x$ (respectively) as follows.

costar link

The link of $x$ separates $x$ from $\mathcal K \setminus x$ . The following lemma generalizes the natural strategy for showing that a connected tree is contractible.

Lemma: If $\mathcal K$ is a geometric simplicial complex and both $\mathcal K \setminus v$ and $\mathcal K / v$ are contractible, then so is $\mathcal K$ .

Instead of proving this lemma, we will work with the more combinatorial notion of collapsibility.

Let $\mathcal K$ be an abstract simplicial complex. A free face of $\mathcal K$ is a non-maximal face which is contained in a unique maximal face. We can collapse $\mathcal K$ to a subcomplex by removing a free face $F$ , along with all faces containing $F$ .

This yeilds an inductive notion of collapsibility: $\mathcal K$ is collapsible if there exists a sequence of collapses that leads from $\mathcal K$ to the empty set. It is straightforward to verify that a geometric simplicial complex $\mathcal K$ is contractible whenever it is collapsible, but the reverse direction does not hold.

Now we state a version of the preceding lemma for collapsibility.

Collapsibility Lemma: If $\mathcal K \setminus v$ and $\mathcal K / v$ are collapsible, then so is $\mathcal K$ .

Proof: The key observation is that the sequence of moves used to collapse $\mathcal K / v$ can be used to collapse $\mathcal K$ to $\mathcal K \setminus v$ (verify using the definition of $\mathcal K / v$ ), at which point we can collapse $\mathcal K \setminus v$ to the empty set by assumption.

Collapsibility and Evasiveness

Before ending the lecture, we should say why collapsibility is relevant to the evasiveness conjecture. The connection is relatively straightforward: Let $f : \{0,1\}^n \to \{0,1\}$ be a non-trivial, monotone boolean function. For a subset $S \subseteq \{1,2,\ldots,n\}$ , let $\chi_S \in \{0,1\}^n$ represent the characteristic vector of $S$ . Then associated to $f$ , we have the simplicial complex

$\mathcal K_{f} = \left\{\vphantom{\bigoplus} S \subseteq \{1,2,\ldots,n\} : f(\chi_S) = 0 \right\}.$

Let $f|_{x_i=0}$ and $f|_{x_i=1}$ be the two functions on $n-1$ bits corresponding to fixing the value of the $i$ th bit. Then we have:

$\displaystyle \mathcal K_{f|_{x_i=0}} = \left\{\vphantom{\bigoplus} S \subseteq \{1,\ldots,i-1,i+1,\ldots,n\} : S \in \mathcal K_f \right\} = \mathcal K_{f} \setminus i,$

$\displaystyle \mathcal K_{f|_{x_i=1}} = \left\{\vphantom{\bigoplus} S \subseteq \{1,\ldots,i-1,i+1,\ldots,n\} : S \cup \{i\} \in \mathcal K_f \right\} = \mathcal K_{f} / i$

A simple induction using the Collapsibility Lemma (which we will do in the next lecture) now yields our main topological connection:

If $f$ is non-evasive, then $\mathcal K_f$ is collapsible (hence also contractible).

[Credits: Some pictures taken from Wikipedia.]

One thought on “Lecture 8a. A primer on simplicial complexes and collapsibility”

Ryan says:

December 21, 2008 at 6:44 am

great article!

tcs math

some mathematics & computation

Lecture 8a. A primer on simplicial complexes and collapsibility

Simplices

Contractibility and collapsibility

Collapsibility and Evasiveness

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