Inmathematics, asimplicial complex is a structuredset ofsimplices (for example,points,line segments,triangles, and theirn-dimensional counterparts) such that all thefaces and intersections of the elements are also included in the set (see illustration). Simplicial complexes should not be confused with the more abstract notion of asimplicial set appearing in modern simplicialhomotopy theory. The purelycombinatorial counterpart to a simplicial complex is anabstract simplicial complex. To distinguish a simplicial complex from an abstract simplicial complex, the former is often called ageometric simplicial complex.^[1]: 7

Asimplicial complex${\displaystyle \mathcal{K}}$ is a set ofsimplices that satisfies the following conditions:

1. Everyface of a simplex from${\displaystyle \mathcal{K}}$ is also in${\displaystyle \mathcal{K}}$.
2. The non-emptyintersection of any two simplices${\displaystyle {\sigma }_1,{\sigma }_2\in \mathcal{K}}$ is a face of both${\displaystyle {\sigma }_1}$ and${\displaystyle {\sigma }_2}$.

See also the definition of anabstract simplicial complex, which loosely speaking is a simplicial complex without an associated geometry.

Asimplicialk-complex${\displaystyle \mathcal{K}}$ is a simplicial complex where the largest dimension of any simplex in${\displaystyle \mathcal{K}}$ equalsk. For instance, a simplicial 2-complex must contain at least one triangle, and must not contain anytetrahedra or higher-dimensional simplices.

Apure orhomogeneous simplicialk-complex${\displaystyle \mathcal{K}}$ is a simplicial complex where every simplex of dimension less thank is a face of some simplex${\displaystyle \sigma \in \mathcal{K}}$ of dimension exactlyk. 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 anon-homogeneous complex is a triangle with a line segment attached to one of its vertices. Pure simplicial complexes can be thought of astriangulations and provide a definition ofpolytopes.

Afacet is a maximal simplex, i.e., any simplex in a complex that isnot a face of any larger simplex.^[2] (Note the difference from a"face" of a simplex). A pure simplicial complex can be thought of as a complex where all facets have the same dimension. For (boundary complexes of)simplicial polytopes this coincides with the meaning from polyhedral combinatorics.

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

For a simplicial complexembedded in ak-dimensional space, thek-faces are sometimes referred to as itscells. The termcell is sometimes used in a broader sense to denote a sethomeomorphic to a simplex, leading to the definition ofcell complex.

Theunderlying space, sometimes called thecarrier of a simplicial complex, is theunion of its simplices. It is usually denoted by${\displaystyle |\mathcal{K}|}$ or${\displaystyle \Vert \mathcal{K}\Vert }$.

Therelative interiors of all simplices in${\displaystyle \mathcal{K}}$ form a partition of its underlying space${\displaystyle |\mathcal{K}|}$: for each point${\displaystyle x\in |\mathcal{K}|}$, there is exactly one simplex in${\displaystyle \mathcal{K}}$ containing${\displaystyle x}$ in its relative interior. This simplex is called thesupport ofx and denoted${\displaystyle \mathrm{supp}(x)}$.^[3]: 9

• 
  Twosimplices and theirclosure.
• 
  Avertex and itsstar.
• 
  Avertex and itslink.

LetK be a simplicial complex and letS be a collection of simplices inK.

Theclosure ofS (denoted${\displaystyle \mathrm{C}\mathrm{l}S}$) is the smallest simplicial subcomplex ofK that contains each simplex inS.${\displaystyle \mathrm{C}\mathrm{l}S}$ is obtained by repeatedly adding toS each face of every simplex inS.

Thestar ofS (denoted${\displaystyle \mathrm{s}\mathrm{t}S}$) is the union of the stars of each simplex inS. For a single simplexs, the star ofs is the set of simplices inK that haves as a face. The star ofS is generally not a simplicial complex itself, so some authors define theclosed star of S (denoted${\displaystyle \mathrm{S}\mathrm{t}S}$) as${\displaystyle \mathrm{C}\mathrm{l}\mathrm{s}\mathrm{t}S}$ the closure of the star of S.

Thelink ofS (denoted${\displaystyle \mathrm{L}\mathrm{k}S}$) equals${\displaystyle \mathrm{C}\mathrm{l}(\mathrm{s}\mathrm{t}(S))\setminus \mathrm{s}\mathrm{t}(\mathrm{C}\mathrm{l}(S))}$. It is the closed star ofS minus the stars of all faces ofS.

Inalgebraic topology, simplicial complexes are often useful for concrete calculations. For the definition ofhomology groups 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 complexes. Infinite complexes are a technical tool basic in algebraic topology. See also the discussion atPolytope of simplicial complexes as subspaces ofEuclidean space made up of subsets, each of which is asimplex. That somewhat more concrete concept is there attributed toAlexandrov. 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, acompacttopological space which is homeomorphic to the geometric realization of a finite simplicial complex is usually called apolyhedron (seeSpanier 1966,Maunder 1996,Hilton & Wylie 1967).

Combinatorialists often study thef-vector of a simplicial d-complex Δ, which is theinteger sequence${\displaystyle (f_0,f_1,f_2,\dots ,f_{d+1})}$, wheref_i is the number of (i−1)-dimensional faces of Δ (by convention,f₀ = 1 unless Δ is the empty complex). For instance, if Δ is the boundary of theoctahedron, then itsf-vector is (1, 6, 12, 8), and if Δ is the first simplicial complex pictured above, itsf-vector is (1, 18, 23, 8, 1). A complete characterization of the possiblef-vectors of simplicial complexes is given by theKruskal–Katona theorem.

By using thef-vector of a simpliciald-complex Δ as coefficients of apolynomial (written in decreasing order of exponents), we obtain thef-polynomial of Δ. In our two examples above, thef-polynomials would be${\displaystyle x^3+6x^2+12x+8}$ and${\displaystyle x^4+18x^3+23x^2+8x+1}$, respectively.

Combinatorists are often quite interested in theh-vector of a simplicial complex Δ, which is the sequence of coefficients of the polynomial that results from pluggingx − 1 into thef-polynomial of Δ. Formally, if we writeF_Δ(x) to mean thef-polynomial of Δ, then theh-polynomial of Δ is

${\displaystyle F_{\mathrm{\Delta }}(x-1)=h_0{x}^{d+1}+h_1{x}^d+h_2{x}^{d-1}+\cdots +h_{d}x+h_{d+1}}$

and theh-vector of Δ is

${\displaystyle (h_0,h_1,h_2,\cdots ,h_{d+1}).}$

We calculate the h-vector of the octahedron boundary (our first example) as follows:

${\displaystyle F(x-1)=(x-1{)}^3+6(x-1{)}^2+12(x-1)+8=x^3+3x^2+3x+1.}$

So theh-vector of the boundary of the octahedron is (1, 3, 3, 1). It is not an accident thish-vector is symmetric. In fact, this happens whenever Δ is the boundary of a simplicialpolytope (these are theDehn–Sommerville equations). In general, however, theh-vector of a simplicial complex is not even necessarily positive. For instance, if we take Δ to be the 2-complex given by two triangles intersecting only at a common vertex, the resultingh-vector is (1, 3, −2).

A complete characterization of all simplicial polytopeh-vectors is given by the celebratedg-theorem ofStanley, Billera, and Lee.

Simplicial complexes can be seen to have the same geometric structure as thecontact graph of asphere packing (a graph where vertices are the centers of spheres and edges exist if the corresponding packing elements touch each other) and as such can be used to determine the combinatorics of sphere packings, such as the number of touching pairs (1-simplices), touching triplets (2-simplices), and touching quadruples (3-simplices) in a sphere packing.

A triangulation of atopological space${\displaystyle X}$ is ahomeomorphism${\displaystyle t:|\mathcal{T}|\to X}$ where${\displaystyle \mathcal{T}}$ is a simplicial complex.

Topological spaces do not necessarily admit a triangulation and if they do, it is never unique.Topological manifolds of dimension${\displaystyle d\le 3}$ are always triangulable,^[4][5][6] but not necessarily for${\displaystyle d>3}$.^[7][8]

Differentiable manifolds of any dimension${\displaystyle d\ge 1}$ admit triangulations.^[9]

Any abstract${\displaystyle d}$-dimensional simplicial complex can be embedded in a${\displaystyle (2d+1)}$-dimensional space.^{[10]: Ch. I Th. 3 [11]: Ch. IV §1.9} This result is piecewise linear counterpart of the (weak)Whitney embedding theorem.

Thesimplicial complex recognition problem is: given a finite simplicial complex, decide whether it is homeomorphic to a given geometric object. This problem isundecidable for anyd-dimensional manifolds for${\displaystyle d\ge 5}$.^{[12][13]: 9–11}

• Abstract simplicial complex
• Barycentric subdivision
• Causal dynamical triangulation
• Delta set
• Loop quantum gravity
• Polygonal chain – 1 dimensional simplicial complex
• Tucker's lemma
• Simplex tree

• Spanier, Edwin H. (1966),Algebraic Topology, Springer,ISBN 0-387-94426-5
• Maunder, Charles R.F. (1996),Algebraic Topology (Reprint of the 1980 ed.), Mineola, NY: Dover,ISBN 0-486-69131-4,MR 1402473
• Hilton, Peter J.;Wylie, Shaun (1967),Homology Theory, New York:Cambridge University Press,ISBN 0-521-09422-4,MR 0115161

• Weisstein, Eric W."Simplicial complex".MathWorld.

• Topological spaces
• Algebraic topology
• Simplicial sets
• Triangulation (geometry)

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