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.


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.



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

  1. If S \in \mathcal K and F is a face of S, then F \in \mathcal K, and
  2. 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.


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

  1. \mathcal K \neq \emptyset
  2. 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

  1. \Phi(\cdot,0) is the identity on T.
  2. \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.

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


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.

After collapsing the free face A.
After collapsing the free face A.

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 ith 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.]

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