It’s most intuitive to start with the geometric viewpoint, in which case an -simplex is defined to be the convex hull of affinely independent points in . These points are called the vertices of the simplex. Here are examples for
A simplicial complex is then a collection of simplices glued together along lower-dimensional simplices. More formally, if is a (geometric) simplex, then a face of is a subset formed by taking the convex hull of a subset of the vertices of .
Finally, a (geometric) simplicial complex is a collection of simplices such that
- If and is a face of , then , and
- If and , then is a face of both and .
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 of vertices, together with a collection of subsets of satisfying the axioms
- If and , then .
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 affinely independent points in and adding in the appropriate convex hulls. (Exercise: Every 1-dimensional complex can be realized in , but not .) 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 is contractible if there exists a continuous mapping such that
- is the identity on .
- for some fixed .
In other words, can be contracted to a point in place. For instance, the closed unit ball in 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
The second, called the link of in is
For example, consider the following complex , in which a distinguished simplex is marked.
Then we form the two simplices and (respectively) as follows.
The link of separates from . The following lemma generalizes the natural strategy for showing that a connected tree is contractible.
Lemma: If is a geometric simplicial complex and both and are contractible, then so is .
Instead of proving this lemma, we will work with the more combinatorial notion of collapsibility.
Let be an abstract simplicial complex. A free face of is a non-maximal face which is contained in a unique maximal face. We can collapse to a subcomplex by removing a free face , along with all faces containing .
This yeilds an inductive notion of collapsibility: is collapsible if there exists a sequence of collapses that leads from to the empty set. It is straightforward to verify that a geometric simplicial complex 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 and are collapsible, then so is .
Proof: The key observation is that the sequence of moves used to collapse can be used to collapse to (verify using the definition of ), at which point we can collapse 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 be a non-trivial, monotone boolean function. For a subset , let represent the characteristic vector of . Then associated to , we have the simplicial complex
Let and be the two functions on bits corresponding to fixing the value of the th bit. Then we have:
A simple induction using the Collapsibility Lemma (which we will do in the next lecture) now yields our main topological connection:
If is non-evasive, then is collapsible (hence also contractible).
[Credits: Some pictures taken from Wikipedia.]