- Simplicial complex
In

mathematics , a**simplicial complex**is atopological space of a particular kind, constructed by "gluing together" points,line segment s,triangle s, and their "n"-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of asimplicial set appearing in modern simplicial homotopy theory.**Definitions**A

**simplicial complex**$mathcal\{K\}$ is a set ofsimplices 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 ofcell complex .The

**underlying space**, sometimes called the**carrier**of a simplicial complex is the union of its simplices.**Closure, star, and link**The

**closure**of a set of simplices "S" (denoted Cl "S") is the smallest simplicial complex containing all the simplices in S. In other words, Cl "S" is the set containing all faces of every simplex in "S".The

**star**of a set of simplices "S" (denoted St "S") with respect to a simplicial complex "K" is the set of all simplices in "K" which have simplices in "S" as faces. (Note that the star is not necessarily a simplicial complex.)The

**link**of a set of simplices "S" (denoted Lk "S") with respect to a simplicial complex "K" equals Cl St "S" - St "S". The link of "S" is in a sense the "boundary" of S with respect to K.**Algebraic topology**In

algebraic topology simplicial complexes are often useful for concrete calculations. For the definition ofhomology group s of a simplicial complex, one can read the correspondingchain complex directly, provided that consistent orientations are made of all simplices. The requirements ofhomotopy theory lead to the use of more general spaces, theCW complex es. Infinite complexes are a technical tool basic inalgebraic topology . See also the discussion atpolytope of simplicial complexes as subspaces of Euclidean space, made up of subsets each of which is asimplex . 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 apolyhedron (see harvnb|Spanier|1966, harvnb|Maunder|1996, harvnb|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.

**References****ee also***

barycentric subdivision

*abstract simplicial complex

*causal dynamical triangulation

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