Simplicial complex

A simplicial 3-complex.

An abstract simplicial complex that is not a valid simplicial complex.

In mathematics, a simplicial complex is a topological space of a particular kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory.

Definitions

A simplicial complex $\mathcal{K}$ is a set of simplices that satisfies the following conditions:

1. Any face of a simplex from $\mathcal{K}$ is also in $\mathcal{K}$.

2. The intersection of any two simplices $\sigma_1, \sigma_2 \in \mathcal{K}$ is a face of both $\sigma_1$ and $\sigma_2$.

Note that the empty set is a face of every simplex. See also the definition of an abstract simplicial complex, which loosely speaking is a simplicial complex without an associated geometry.

A simplicial $k$-complex $\mathcal{K}$ is a simplicial complex where the largest dimension of any simplex in $\mathcal{K}$ equals k. For instance, a simplicial 2-complex must contain at least one triangle, and must not contain any tetrahedra or higher-dimension simplices.

A pure or homogeneous simplicial k-complex $\mathcal{K}$ is a simplicial complex where every simplex of dimension less than k is the face of some simplex $\sigma \in \mathcal{K}$ of dimension exactly k. Informally, a pure 1-complex "looks" like it's made of a bunch of lines, a 2-complex "looks" like it's made of a bunch of triangles, etc. An example of a non-homogeneous complex is a triangle with a line segment attached to one of its vertices.

A facet is any simplex in a complex that is not the face of any larger simplex. (Note the difference from the "facet" of a simplex.) A pure simplicial complex can be thought of as a complex where all facets have the same dimension.

Sometimes the term face is used to refer to a simplex of a complex, not to be confused with the face of a simplex.

For a simplicial complex embedded in a k-dimensional space, the k-faces are sometimes referred to as its cells. The term cell is sometimes used in a broader sense to denote a set homeomorphic to a simplex, leading to the definition of cell complex.

The underlying space, sometimes called the carrier of a simplicial complex is the union of its simplices.

Closure, star, and link

Two simplices and their closure.
A simplex and its star.	A simplex and its link.

Let K be a simplicial complex and let S be a collection of simplicies in K.

The closure of S (denoted Cl S) is the smallest simplicial subcomplex of K that contains each simplex in S. Cl S is obtained by adding to S each face of every simplex in S.

The star of S (denoted St S) is the set of all simplices in K that have faces in S. (Note that the star is not necessarily a simplicial complex.)

The link of S (denoted Lk S) equals Cl St S - St S. One may regard the link of S as a "boundary" of S with respect to K.

Algebraic topology

In algebraic topology simplicial complexes are often useful for concrete calculations. For the definition of homology groups of a simplicial complex, one can read the corresponding chain complex directly, provided that consistent orientations are made of all simplices. The requirements of homotopy theory lead to the use of more general spaces, the CW complexes. Infinite complexes are a technical tool basic in algebraic topology. See also the discussion at polytope of simplicial complexes as subspaces of Euclidean space, made up of subsets each of which is a simplex. That somewhat more concrete concept is there attributed to Alexandrov. Any finite simplicial complex in the sense talked about here can be embedded as a polytope in that sense, in some large number of dimensions. In algebraic topology a compact topological space which is homeomorphic to a the geometric realization of a finite simplicial complex is usually called a polyhedron (see Spanier 1966, Maunder 1996, Hilton & Wylie 1967).

Combinatorics

Combinatorists often study the f-vector of a simplicial d-complex $\Delta$, which is the integral sequence $(f_0, f_1, f_2, ..., f_{d+1})$, where $f$_i is the number of $(i-1)$-dimensional faces of $\Delta$ (by convention, $f$₀$=1$ unless $\Delta$ is the empty complex). For instance, if $\Delta$ is the boundary of the octahedron, then its f-vector is (1, 6, 12, 8), and if $\Delta$ is the first simplicial complex pictured above, its f-vector is (1, 18, 23, 8, 1). A complete characterization of the possible f-vectors of simplicial complexes is given by the Kruskal-Katona theorem.

By using the f-vector of a simplicial d-complex $\Delta$ as coefficients of a polynomial (written in decreasing order of exponents), we obtain the f-polynomial of $\Delta$. In our two examples above, the f-polynomials would be $x^3+6x^2+12x+8$ and $x^4+18x^3+23x^2+8x+1$, respectively.

Combinatorists are often quite interested in the h-vector of a simplicial complex $\Delta$, which is the sequence of coefficients of the polynomial that results from plugging $x-1$ into the f-polynomial of $\Delta$. Formally, if we write $F_\Delta (x)$ to mean the f-polynomial of $\Delta$, then the h-polynomial of $\Delta$ is

$F_\Delta(x-1)=h_0x^{d+1}+h_1x^d+h_2x^{d-1}+...+h_dx+h_{d+1}\,$

and the h-vector of $\Delta$ is $(h_0, h_1, h_2, ..., h_{d+1})$. We calculate the h-vector of the octahedron boundary (our first example) as follows:

$F(x-1)=(x-1)^3+6(x-1)^2+12(x-1)+8=x^3+3x^2+3x+1\,$

So the h-vector of the boundary of the octahedron is (1,3,3,1). It is not an accident this h-vector is symmetric. In fact, this happens whenever $\Delta$ is the boundary of a simplicial polytope (these are the Dehn-Sommerville equations). In general, however, the h-vector of a simplicial complex is not even necessarily positive. For instance, if we take $\Delta$ to be the 2-complex given by two triangles intersecting only at a common vertex, the resulting h-vector is (1,3,-2).

A complete characterization of all simplicial polytope h-vectors is given by the celebrated g-theorem of Stanley, Billera, and Lee.

See also

• Barycentric subdivision
• Abstract simplicial complex
• Causal dynamical triangulation
• Polygonal chain - 1 dimensional simplicial complex

References

• Spanier, E.H. (1966), Algebraic Topology, Springer, ISBN 0-387-94426-5 
• Maunder, C.R.F. (1996), Algebraic Topology, Dover, ISBN 0-486-69131-4 
• Hilton, P.J.; Wylie, S. (1967), Homology Theory, Cambridge University Press, ISBN 0-521-09422-4