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Abstract simplicial complex

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Recall that a simplicial complex is a cell complex with extra conditions:

  1. each of its cells has the structure of a simplex,
  2. the complex contains all faces of each simplex,
  3. two simplices can only share a face.

The last two can be re-written as:

  • if $\tau \in K$ and $\sigma < \tau$ then $\sigma \in K$;
  • if $\tau, \sigma \in K$ then $\tau \cap \sigma < \tau$.

This allows us to represent simplicial complexes in an entirely abstract way.

Let's consider this example.

Example of simplicial complex.png

The complex has:

  • $0$-cells: $A, B, C, D$;
  • $1$-cells: $AB, BC, CA, BD, DC$;
  • $2$-cells: $ABC$.

Observe that this is just a list of subsets of set $S = \{A,B,C,D \}$! This list captures everything about the complex and can be used as a sole source of its reconstruction.

Not every list of subsets of $S$ would work though. Consider $\{AB, BC \}$ for example. This can't be a complex because the faces of these $1$-simplices are absent. However, once we add those to the list, the first condition of simplicial complex above is satisfied and the second too, automatically. Indeed, the list is: $$AB, BC, A, B, C.$$

Turns out the first condition is the only one we have to verify.

Definition. A collection $K$ of subsets of set $S$ is called an abstract simplicial complex if all subsets of any element of $K$ are also elements of $K$, i.e.,

if $\tau \in K$ and $\sigma < \tau$ then $\sigma \in K$.

In this sense, what is an abstract $n$-simplex? It's simply any finite set $A_0A_1...A_n$!

A more economic way of presenting of a simplicial complex is to list only the largest simplices and then use this condition to recover the rest. In the example above we only need to list: $$ABC, BD, DC.$$

Example. The realization of the complex $$ABD, BCD, ACD, ABE, BCE, ACE$$ is found by putting up the vertices first and then adding these $6$ triangles one by one. The result is this;

Realization of abstract simplicial complex.jpg

It's homeomorphic to the sphere.

See also Triangulations of surfaces.

What about topological spaces? Can we represent them as simplicial complexes? Yes, via the nerve of cover construction.