# Simplicial Complex

A simplicial complex is a set of simplices that satisfies the following conditions:

1. Any face of a simplex from is also in .
2. The intersection of any two simplices is a face of both and .

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 is a simplicial complex where the largest dimension of any simplex in 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 is a simplicial complex where every simplex of dimension less than k is a face of some simplex 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 a face of any larger simplex. (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.

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

### Other articles related to "simplicial complex, simplicial, complex":

Simplicial Complex - Combinatorics
... Combinatorialists often study the f-vector of a simplicial d-complex Δ, which is the integral sequence, where fi is the number of (i−1)-dimensional ... of the octahedron, then its f-vector is (1, 6, 12, 8), and if Δ 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 ...
Simplicial Homology
... In mathematics, in the area of algebraic topology, simplicial homology is a theory with a finitary definition, and is probably the most tangible variant of homology ... Simplicial homology concerns topological spaces whose building blocks are n-simplexes, the n-dimensional analogs of triangles ... By definition, such a space is homeomorphic to a simplicial complex (more precisely, the geometric realization of an abstract simplicial complex) ...
Collapse (topology) - Definition
... Let K be a simplicial complex, and suppose that s is a simplex in K ... If we remove s and t from K, we obtain another simplicial complex, which we call an elementary collapse of K ... A simplicial complex that has a collapse to a point, implying all other points were in free pairs, is called collapsible ...
Homotopy Groups Of Spheres - Computational Methods
... If X is any finite simplicial complex with finite fundamental group, in particular if X is a sphere of dimension at least 2, then its homotopy groups are all finitely generated ... computing all homotopy groups of any finite simply connected simplicial complex, but in practice it is too cumbersome to use for computing anything other than the first few ... ordinary cohomology mod p with a generalized cohomology theory, such as complex cobordism or, more usually, a piece of it called Brown–Peterson ...
Homological Algebra - Chain Complexes and Homology
... The chain complex is the central notion of homological algebra ... Every chain complex defines two further sequences of abelian groups, the cycles Zn = Ker dn and the boundaries Bn = Im dn+1, where Ker d and Im d denote the kernel and the ... group Hn(C) as the factor group of the n-cycles by the n-boundaries, A chain complex is called acyclic or an exact sequence if all its homology groups are zero ...

### Famous quotes containing the word complex:

All of life and human relations have become so incomprehensibly complex that, when you think about it, it becomes terrifying and your heart stands still.
Anton Pavlovich Chekhov (1860–1904)