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Abstract simplicial complex
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Recall that a simplicial complex is a cell complex with extra conditions:
- each of its cells has the structure of a simplex,
- the complex contains all faces of each simplex,
- 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.
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.,
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;
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.