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.
A simplicial complex is a set of simplices that satisfies the following conditions:
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-dimensional 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.
Let K be a simplicial complex and let S be a collection of simplices 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 repeatedly adding to S each face of every simplex in S.
The star of S (denoted St S) is the union of the stars of each simplex in S. For a single simplex s, the star of s is the set of simplices having s as a face. (Note that the star of S is generally not a simplicial complex itself).
The link of S (denoted Lk S) equals Cl St S − St Cl S. It is the closed star of S minus the stars of all faces of S.
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 the geometric realization of a finite simplicial complex is usually called a polyhedron (see Spanier 1966, Maunder 1996, Hilton & Wylie 1967).
Combinatorialists often study the f-vector of a simplicial d-complex Δ, which is the integer sequence , where fi 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 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.
By using the f-vector of a simplicial d-complex Δ as coefficients of a polynomial (written in decreasing order of exponents), we obtain the f-polynomial of Δ. In our two examples above, the f-polynomials would be and , respectively.
Combinatorists are often quite interested in the h-vector of a simplicial complex Δ, which is the sequence of coefficients of the polynomial that results from plugging x − 1 into the f-polynomial of Δ. Formally, if we write FΔ(x) to mean the f-polynomial of Δ, then the h-polynomial of Δ is
and the h-vector of Δ is
We calculate the h-vector of the octahedron boundary (our first example) as follows:
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 Δ 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 Δ 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.
Simplicial complexes can be seen to have the same geometric structure as the contact graph of a sphere 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.
In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given space.
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, to other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topology. Similar constructions are available in a wide variety of other contexts, such as abstract algebra, groups, Lie algebras, Galois theory, and algebraic geometry.
In mathematics, a building is a combinatorial and geometric structure which simultaneously generalizes certain aspects of flag manifolds, finite projective planes, and Riemannian symmetric spaces. They were initially introduced by Jacques Tits as a means to understand the structure of exceptional groups of Lie type. The more specialized theory of Bruhat–Tits buildings plays a role in the study of p-adic Lie groups analogous to that of the theory of symmetric spaces in the theory of Lie groups.
In algebraic topology, a branch of mathematics, singular homology refers to the study of a certain set of algebraic invariants of a topological space X, the so-called homology groups Intuitively, singular homology counts, for each dimension n, the n-dimensional holes of a space. Singular homology is a particular example of a homology theory, which has now grown to be a rather broad collection of theories. Of the various theories, it is perhaps one of the simpler ones to understand, being built on fairly concrete constructions.
In geometry, the barycentric subdivision is a standard way of dividing an arbitrary convex polygon into triangles, a convex polyhedron into tetrahedra, or, in general, a convex polytope into simplices with the same dimension, by connecting the barycenters of their faces in a specific way.
In combinatorics, an abstract simplicial complex (ASC) is a family of sets that is closed under taking subsets, i.e., every subset of a set in the family is also in the family. It is a purely combinatorial description of the geometric notion of a simplicial complex. For example, in a 2-dimensional simplicial complex, the sets in the family are the triangles, their edges, and their vertices.
In mathematics, specifically algebraic topology, Čech cohomology is a cohomology theory based on the intersection properties of open covers of a topological space. It is named for the mathematician Eduard Čech.
In mathematics, a simplicial set is an object made up of "simplices" in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined as a contravariant functor from the simplex category to the category of sets. Simplicial sets were introduced in 1950 by Samuel Eilenberg and J. A. Zilber.
In algebraic topology, simplicial homology is the sequence of homology groups of a simplicial complex. It formalizes the idea of the number of holes of a given dimension in the complex. This generalizes the number of connected components.
In mathematics, particularly in algebraic topology, the n-skeleton of a topological space X presented as a simplicial complex refers to the subspace Xn that is the union of the simplices of X of dimensions m ≤ n. In other words, given an inductive definition of a complex, the n-skeleton is obtained by stopping at the n-th step.
In mathematics, Kan complexes and Kan fibrations are part of the theory of simplicial sets. Kan fibrations are the fibrations of the standard model category structure on simplicial sets and are therefore of fundamental importance. Kan complexes are the fibrant objects in this model category. The name is in honor of Daniel Kan.
In mathematics, Hochschild homology is a homology theory for associative algebras over rings. There is also a theory for Hochschild homology of certain functors. Hochschild cohomology was introduced by Gerhard Hochschild (1945) for algebras over a field, and extended to algebras over more general rings by Henri Cartan and Samuel Eilenberg (1956).
In algebraic combinatorics, the h-vector of a simplicial polytope is a fundamental invariant of the polytope which encodes the number of faces of different dimensions and allows one to express the Dehn–Sommerville equations in a particularly simple form. A characterization of the set of h-vectors of simplicial polytopes was conjectured by Peter McMullen and proved by Lou Billera and Carl W. Lee and Richard Stanley (g-theorem). The definition of h-vector applies to arbitrary abstract simplicial complexes. The g-conjecture stated that for simplicial spheres, all possible h-vectors occur already among the h-vectors of the boundaries of convex simplicial polytopes. It was proven in December 2018 by Karim Adiprasito.
In algebraic topology, a discipline within mathematics, the acyclic models theorem can be used to show that two homology theories are isomorphic. The theorem was developed by topologists Samuel Eilenberg and Saunders MacLane. They discovered that, when topologists were writing proofs to establish equivalence of various homology theories, there were numerous similarities in the processes. Eilenberg and MacLane then discovered the theorem to generalize this process.
In mathematics, a Δ-setS, often called a semi-simplicial set, is a combinatorial object that is useful in the construction and triangulation of topological spaces, and also in the computation of related algebraic invariants of such spaces. A Δ-set is somewhat more general than a simplicial complex, yet not quite as general as a simplicial set.
In mathematics, a Stanley–Reisner ring, or face ring, is a quotient of a polynomial algebra over a field by a square-free monomial ideal. Such ideals are described more geometrically in terms of finite simplicial complexes. The Stanley–Reisner ring construction is a basic tool within algebraic combinatorics and combinatorial commutative algebra. Its properties were investigated by Richard Stanley, Melvin Hochster, and Gerald Reisner in the early 1970s.
In mathematics, a pseudomanifold is a special type of topological space. It looks like a manifold at most of its points, but it may contain singularities. For example, the cone of solutions of forms a pseudomanifold.
In mathematics, a shelling of a simplicial complex is a way of gluing it together from its maximal simplices in a well-behaved way. A complex admitting a shelling is called shellable.
In mathematics, a polyhedral complex is a set of polyhedra in a real vector space that fit together in a specific way. Polyhedral complexes generalize simplicial complexes and arise in various areas of polyhedral geometry, such as tropical geometry, splines and hyperplane arrangements.
Discrete calculus or the calculus of discrete functions, is the mathematical study of incremental change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. The word calculus is a Latin word, meaning originally "small pebble"; as such pebbles were used for calculation, the meaning of the word has evolved and today usually means a method of computation. Meanwhile, calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the study of continuous change.