Mathematical set composed of points, line segments, triangles, and their n-dimensional counterparts
A simplicial 3-complex.
In mathematics, a simplicial complex is a set composed of 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. The purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex. To distinguish a simplicial complex from an abstract simplicial complex, the former is often called a geometric simplicial complex.[1]: 7
^Matoušek, Jiří (2007). Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry (2nd ed.). Berlin-Heidelberg: Springer-Verlag. ISBN 978-3-540-00362-5. Written in cooperation with Anders Björner and Günter M. Ziegler
, Section 4.3
and 20 Related for: Simplicial complex information
counterpart to a simplicialcomplex is an abstract simplicialcomplex. To distinguish a simplicialcomplex from an abstract simplicialcomplex, the former...
In combinatorics, an abstract simplicialcomplex (ASC), often called an abstract complex or just a complex, is a family of sets that is closed under taking...
In algebraic topology, simplicial homology is the sequence of homology groups of a simplicialcomplex. It formalizes the idea of the number of holes of...
operation on simplicialcomplexes. In algebraic topology it is sometimes useful to replace the original spaces with simplicialcomplexes via triangulations:...
In mathematics, a simplicial set is an object composed of simplices in a specific way. Simplicial sets are higher-dimensional generalizations of directed...
a simplicialcomplex, its fundamental group can be described explicitly in terms of generators and relations. If X is a connected simplicialcomplex, an...
illustration). Simplicialcomplexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory...
Coxeter complex, named after H. S. M. Coxeter, is a geometrical structure (a simplicialcomplex) associated to a Coxeter group. Coxeter complexes are the...
illustration). Simplicialcomplexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory...
The simplicialcomplex recognition problem is a computational problem in algebraic topology. Given a simplicialcomplex, the problem is to decide whether...
models like simplicial and cell complexes. Combinatorial Complexes : Generalize and bridge the gaps between simplicialcomplexes, cell complexes, and hypergraphs...
between spaces that are built up from simplices—that is, finite simplicialcomplexes. The general continuous mapping between such spaces can be represented...
for each vertex i of X ', a simplicialcomplex Li' endowed with a rigid simplicial action of a finite group Γi. a simplicial map φi of Li' onto the link...
In mathematics, a Δ-set, often called a Δ-complex or a semi-simplicial set, is a combinatorial object that is useful in the construction and triangulation...
n-dimensional simplicialcomplexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicialcomplexes or CW complexes), the...
represented as a simplicialcomplex. A distance function on the underlying space corresponds to a filtration of the simplicialcomplex, that is a nested...
together" simplices to form a simplicialcomplex. The associated combinatorial structure is called an abstract simplicialcomplex, in which context the word...
continuous maps from the standard n-simplex into X; if K is a simplicialcomplex then the simplicial chains Cn(K) are formal linear combinations of the n-simplices...
properties than simplicialcomplexes, but still retains a combinatorial nature that allows for computation (often with a much smaller complex). The C stands...
by 0. An example is the chain complex defining the simplicial homology of a finite simplicialcomplex. A chain complex is bounded above if all modules...
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