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combinatorics, a simplicial (or combinatorial) d-sphere is a simplicial complex homeomorphic to the d-dimensional sphere. Some simplicial spheres arise as the
Simplicial_sphere
Application of homotopy to algebraic varieties
addition a cone of a simplicial (pre)sheaf and a cone of a morphism, but defining these requires the definition of the simplicial spheres. From the fact we
A¹_homotopy_theory
German mathematician
manifolds. In December 2018, he proved Peter McMullen's g-conjecture for simplicial spheres. For his work, he won the 2020 EMS Prize of the European Mathematical
Karim_Adiprasito
Type of mathematical set
In mathematics, a simplicial complex is a structured set of simplices (for example, points, line segments, triangles, and their n-dimensional counterparts)
Simplicial_complex
h-vector applies to arbitrary abstract simplicial complexes. The g-conjecture stated that for simplicial spheres, all possible h-vectors occur already
H-vector
Mathematical construction used in homotopy theory
mathematics, a simplicial set is a sequence of sets with internal order structure (abstract simplices) and maps between them. Simplicial sets are higher-dimensional
Simplicial_set
Topological manifold whose homology coincides with that of a sphere
5-sphere, but its triangulation (induced by some triangulation of A) is not a PL manifold. In other words, this gives an example of a finite simplicial
Homology_sphere
Convex hull of points on moment curve
polytope C(n,d) maximizes the number fi of i-dimensional faces among all simplicial spheres of dimension d − 1 with n vertices. The moment curve in R d {\displaystyle
Cyclic_polytope
How spheres of various dimensions can wrap around each other
Hopf elements. 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
Homotopy_groups_of_spheres
upper bound theorem states that if Δ {\displaystyle \Delta } is a simplicial sphere of dimension d − 1 {\displaystyle d-1} with n {\displaystyle n} vertices
Upper_bound_theorem
Mathematical ring
Richard Stanley found a way to prove the Upper Bound Conjecture for simplicial spheres, which was open at the time, using the face ring construction and
Stanley–Reisner_ring
Computational problem in algebraic topology
The simplicial complex recognition problem is a computational problem in algebraic topology. Given a simplicial complex, the problem is to decide whether
Simplicial complex recognition problem
Simplicial_complex_recognition_problem
g-conjecture on the possible numbers of faces of different dimensions in a simplicial sphere (also Grünbaum conjecture, several conjectures of Kühnel). (Karim
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Representation of mathematical space
mathematics, triangulation describes the replacement of topological spaces with simplicial complexes by the choice of an appropriate homeomorphism. A space that
Triangulation_(topology)
Algebraic structure associated with a topological space
which make the task easier. The simplicial homology groups Hn(X) of a simplicial complex X are defined using the simplicial chain complex C(X), with Cn(X)
Homology_(mathematics)
Combinitorics of Polyhedra
Combinatorial commutative algebra Matroid polytope Order polytope Simplicial sphere Stable matching polytope Ziegler (1995), p. 51. Ziegler (1995), pp
Polyhedral_combinatorics
Mathematical group of the homotopy classes of loops in a topological space
covering space of a finite connected simplicial complex X {\displaystyle X} can also be described directly as a simplicial complex using edge-paths. Its vertices
Fundamental_group
Field of mathematics using techniques from combinatorics and commutative algebra
question was the extension of this characterization from simplicial polytopes to simplicial spheres, the g-conjecture, which was resolved in 2018 by Karim
Combinatorial commutative algebra
Combinatorial_commutative_algebra
Branch of mathematics
illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory
Algebraic_topology
The link in a simplicial complex is a generalization of the neighborhood of a vertex in a graph. The link of a vertex encodes information about the local
Link_(simplicial_complex)
Branch of mathematics
graded chain complexes over a fixed base ring. A simplicial set is an abstract generalization of a simplicial complex and can play a role of a "space" in some
Homotopy_theory
expresses the fact that Euler characteristic of a (d − 1)-dimensional simplicial sphere is equal to 1 + (−1)d − 1. Dehn–Sommerville equations with different
Dehn–Sommerville_equations
Multi-dimensional generalization of triangle
building blocks of discretizations of spacetime; that is, to build simplicial manifolds. 3-sphere Aitchison geometry Causal dynamical triangulation Complete graph
Simplex
British mathematician
Carl W. Lee, and Richard P. Stanley, characterizing the f-vectors of simplicial spheres. The McMullen problem is an unsolved question in discrete geometry
Peter_McMullen
Topological invariant in mathematics
(When only triangular faces are used, they are two-dimensional finite simplicial complexes.) In general, for any finite CW-complex, the Euler characteristic
Euler_characteristic
Topological space that locally resembles Euclidean space
discretization of a manifold. It usually means a piecewise linear manifold made by simplicial complexes. A digital manifold is a special kind of combinatorial manifold
Manifold
Branch of geometry that studies combinatorial properties and constructive methods
illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory
Discrete_geometry
Mathematics glossary
definition of a spectrum. A simplicial set is not thought of as a space; i.e., we generally distinguish between simplicial sets and their geometric realizations
Glossary of algebraic topology
Glossary_of_algebraic_topology
Yugoslav American mathematician (1929-2018)
Elongated square gyrobicupola Goldner–Harary graph Pentagram map Simplicial sphere Star coloring Star polygon Grünbaum's theorem Grünbaum–Rigby configuration
Branko_Grünbaum
Tiling of n-dimensional space
In geometry, the simplicial honeycomb (or n-simplex honeycomb) is a dimensional infinite series of honeycombs, based on the A ~ n {\displaystyle {\tilde
Simplicial_honeycomb
Mathematical theory
L)&\cong 0\\\end{aligned}}} Let X {\displaystyle X} be an abstract simplicial complex on a vertex set V {\displaystyle V} of size n {\displaystyle n}
Alexander_duality
Topological property
the k-th homology group of a simplicial complex depends only on the simplices of dimension at most k+1 (see simplicial homology). Therefore, the above
Homotopical_connectivity
Concepts in algebraic topology
composition. This creates a technical problem which can be solved using simplicial techniques: giving a method for constructing a model for homotopy colimits
Homotopy_colimit_and_limit
Topological manifold with a piecewise linear structure on it
space. See digital topology. Simplicial manifold A PL structure also requires that the link of a simplex be a PL-sphere. An example of a topological triangulation
Piecewise_linear_manifold
Branch of mathematics
topological data analysis is to: Replace a set of data points with a family of simplicial complexes, indexed by a proximity parameter. Analyse these topological
Topology
Map between simplicial sets with lifting property
part of the theory of simplicial sets. Kan fibrations are the fibrations of the standard model category structure on simplicial sets and are therefore
Kan_fibration
Operation in topology
homeomorphic to the n-dimensional sphere S n {\displaystyle S^{n}} . The n-fold k-wise deleted join of a simplicial complex A is defined as: A Δ ( k )
Join_(topology)
Construction for n-dimensional noise functions
An implementation typically involves four steps: coordinate skewing, simplicial subdivision, gradient selection, and kernel summation. An input coordinate
Simplex_noise
Complex recording the pattern of intersections between a topological family's sets
{\displaystyle N(C)} , making N ( C ) {\displaystyle N(C)} an abstract simplicial complex. Hence N(C) is often called the nerve complex of C {\displaystyle
Nerve_complex
1967 mathematics textbook
structures, the proof of the g {\displaystyle g} -conjecture for simplicial spheres, and Kalai's 3d conjecture. The second edition also provides an improved
Convex_Polytopes
Gives a homomorphism from homotopy groups to homology groups
Hurewicz theorem for topological spaces can also be stated for n-connected simplicial sets satisfying the Kan condition. Rational Hurewicz theorem: Let X be
Hurewicz_theorem
W)\to \operatorname {map} (X,W)} is required to be a weak equivalence (of simplicial sets) for any C-local object W. An object W is called C-local if it is
Bousfield_localization
Shape in hyperbolic geometry
the triakis tetrahedron is simplicial and non-ideal, and the 4-regular non-ideal example above shows that for non-simplicial polyhedra, having all degrees
Ideal_polyhedron
Point where two or more curves, lines, or edges meet
of the polytope, and in that a graph can be viewed as a 1-dimensional simplicial complex the vertices of which are the graph's vertices. However, in graph
Vertex_(geometry)
Algebraic topology uses abstract algebra to study topological spaces
Simplex Simplicial complex Polytope Triangulation Barycentric subdivision Simplicial approximation theorem Abstract simplicial complex Simplicial set Simplicial
List of algebraic topology topics
List_of_algebraic_topology_topics
Line constructed from a triangle
circumcenter along this line. The center of the twelve-point sphere also lies on the Euler line. A simplicial polytope is a polytope whose facets are all simplices
Euler_line
Two pentagonal pyramids fused base-to-base
Regardless of any type of its triangular faces, the pentagonal bipyramid is a simplicial polyhedron like any other bipyramid. The vertices and edges of a pentagonal
Pentagonal_bipyramid
Conjecture in knot theory relating quantum invariants and hyperbolic geometry
\operatorname {vol} (S^{3}\backslash K)} is the simplicial volume of the complement of K {\displaystyle K} in the 3-sphere, defined as follows. By the JSJ decomposition
Volume_conjecture
Establishes the concept of stabilization of homotopy groups
Let X be an n-connected pointed space (a pointed CW-complex or pointed simplicial set). The map X → Ω ( Σ X ) {\displaystyle X\to \Omega (\Sigma X)} induces
Freudenthal suspension theorem
Freudenthal_suspension_theorem
Subdivision of the plane by lines
There are three known infinite families of simplicial arrangements, as well as many sporadic simplicial arrangements that do not fit into any known family
Arrangement_of_lines
Generalized manifold
for each vertex i of X ', a simplicial complex Li' endowed with a rigid simplicial action of a finite group Γi. a simplicial map φi of Li' onto the link
Orbifold
Solid with eight equal triangular faces
regular octahedron is an example of many classifications as deltahedron and simplicial polyhedron. Regular octahedra occur in nature and science, such as the
Regular_octahedron
example, for nice spaces—say, simplicial complexes—that any continuous map of X to the circle or to the higher spheres is null-homotopic. If a space X
Acyclic_space
Abstract simplicial complex describing a graph's cliques
graph. The clique complex X(G) of an undirected graph G is an abstract simplicial complex (that is, a family of finite sets closed under the operation of
Clique_complex
Mathematical structure
building of a group of Lie type is the same as that of a bouquet of spheres. The simplicial structure of the affine and spherical buildings associated to SLn(Qp)
Building_(mathematics)
Two tetrahedra joined by one face
of its triangular faces with any type, the triangular bipyramid is a simplicial polyhedron like other infinitely many bipyramids. A right bipyramid is
Triangular_bipyramid
Algebraic construct classifying topological spaces
model categories. It is possible to define abstract homotopy groups for simplicial sets. Homology groups are similar to homotopy groups in that they can
Homotopy_group
Type of topological space
dimensions in specific ways. The notion generalizes both manifolds and simplicial complexes and has particular significance for algebraic topology. It was
CW_complex
Simplicial complex
complex, named after H. S. M. Coxeter, is a geometrical structure (a simplicial complex) associated to a Coxeter group. Coxeter complexes are the basic
Coxeter_complex
Concept in homotopy theory
all projective. The category SSet {\displaystyle {\textbf {SSet}}} of simplicial setspg 1.3 there is a model category structure where the fibrations are
Cofibration
Shape with three sides
as the simplex, and the polytopes with triangular facets known as the simplicial polytopes. Each triangle has many special points inside it, on its edges
Triangle
Algebraic tool for computing topological spaces' invariants
sequence holds for a variety of cohomology and homology theories, including simplicial homology and singular cohomology. In general, the sequence holds for those
Mayer–Vietoris_sequence
Two-dimensional manifold
as the boundaries of three-dimensional solid figures; for example, the sphere is the boundary of the solid ball. Other surfaces arise as graphs of functions
Surface_(topology)
Mathematical theory
group), while for i ≥ 1 we have Hi(P) = {0}. More generally if X is a simplicial complex or finite CW complex, then the group H0(X) is the free abelian
Reduced_homology
Convex polyhedron projected from hypercube
zonohedron corresponds in this way to a simplicial arrangement, one in which each face is a triangle. Simplicial arrangements of great circles correspond
Zonohedron
Branch of the mathematical field of graph theory
a short circuit. To an undirected graph we may associate an abstract simplicial complex C with a single-element set per vertex and a two-element set per
Topological_graph_theory
Tool in homological algebra
right by 0. An example is the chain complex defining the simplicial homology of a finite simplicial complex. A chain complex is bounded above if all modules
Chain_complex
This means that one can consider symmetric products of objects like simplicial sets as well. Moreover, if the category is cartesian closed, the distributive
Symmetric_product_(topology)
The odd–even condition follows from Euler's formula. Any simplicial generalized homology sphere is an Eulerian lattice. Let L be a regular cell complex
Eulerian_poset
Mathematical theories
respect to dimension, for extending a continuous mapping defined on a simplicial complex, or CW complex. It is traditionally called Eilenberg obstruction
Obstruction_theory
Topological manifold in mathematics
triangulable as a simplicial complex. The manifold can be constructed by first plumbing together disc bundles of Euler number 2 over the sphere, according to
E8_manifold
Branch of topology
homeomorphic to any simplicial complex. In dimension at least 5 the existence of topological manifolds not homeomorphic to a simplicial complex was an open
Low-dimensional_topology
Non-orientable surface with one edge
come from an abstract simplicial complex, because all three triangles share the same three vertices, while abstract simplicial complexes require each
Möbius_strip
co-dimension 0 subcomplex C ⊂ N {\displaystyle C\subset N} together with a simplicial isomorphism ϕ : C → C ′ ⊂ ∂ Δ n + 1 {\displaystyle \phi :C\to C'\subset
Pachner_moves
Triangulation method
developed. Typically, the domain to be meshed is specified as a coarse simplicial complex; for the mesh to be numerically stable, it must be refined, for
Delaunay_triangulation
Decomposition of a manifold into standard pieces
3-manifolds. Handlebodies play a similar role in the study of manifolds as simplicial complexes and CW complexes play in homotopy theory, allowing one to analyze
Handlebody
Result in algebraic topology
Let M be a finite simplicial complex of dimension dim(M) = m ≥ 5. Suppose that M has the homotopy type of the m-dimensional sphere Sm and that M is locally
Stallings–Zeeman_theorem
Mathematical concept
In mathematics, a shelling of a simplicial complex is a way of gluing it together from its maximal simplices (simplices that are not a face of another
Shelling_(topology)
Convex polyhedron with 12 triangular faces
vertices may be placed on a sphere and can also be used as a minimum possible Lennard-Jones potential among all eight-sphere clusters. The snub disphenoid
Snub_disphenoid
Mathematical object
sequence X n {\displaystyle X_{n}} of pointed topological spaces or pointed simplicial sets together with the structure maps S 1 ∧ X n → X n + 1 {\displaystyle
Spectrum_(topology)
Topological space with only one nontrivial homotopy group
to use the geometric realization of simplicial abelian groups. This gives an explicit presentation of simplicial abelian groups which represent Eilenberg–MacLane
Eilenberg–MacLane_space
Spectral sequence in algebraic topology
Inventiones Mathematicae 3 (1967), 172–178, EuDML. The case of simplicial sets is treated in Paul Goerss, Rick Jardine, Simplicial homotopy theory, Birkhäuser
Serre_spectral_sequence
Theory for associative algebras over rings
family of modules ( C n ( A , M ) , b ) {\displaystyle (C_{n}(A,M),b)} a simplicial object in the category of k-modules, i.e., a functor Δo → k-mod, where
Hochschild_homology
homotopy theory using the language of univalent foundations to reason about simplicial sets and cubical sets. Synthetic computability theory develops computability
Synthetic_mathematics
Steinitz is related to the notion of an abstract simplicial complex and it differs from a simplicial complex by the property that its elements are not
Abstract_cell_complex
Aspect of algebraic topology
generalized in several different directions (group actions, foliations, simplicial complexes, etc.). Ganea conjecture Systolic category Ralph H. Fox, On
Lusternik–Schnirelmann category
Lusternik–Schnirelmann_category
Describes the fundamental group in terms of a cover by two open path-connected subspaces
Allegretti, Simplicial Sets and Van Kampen's Theorem (Discusses generalized versions of Van Kampen's theorem applied to topological spaces and simplicial sets)
Seifert–Van_Kampen_theorem
Simplex formed from a right-angled path
group. This is a barycentric subdivision. We proceed to describe the "simplicial subdivision" of a regular polytope, beginning with the one-dimensional
Schläfli_orthoscheme
Convex hull of a finite set of points in a Euclidean space
e. as a spherical tiling. A convex polytope can be decomposed into a simplicial complex, or union of simplices, satisfying certain properties. Given a
Convex_polytope
Area of mathematics
Instead of smooth curves and surfaces, there are polygons, meshes, and simplicial complexes. It is used in the study of computer graphics, geometry processing
Discrete differential geometry
Discrete_differential_geometry
American mathematician (born 1931)
day. In 1961, Milnor disproved the Hauptvermutung by illustrating two simplicial complexes that are homeomorphic but combinatorially distinct, using the
John_Milnor
Concept in algebra
Q-construction, which produces a topological space, the S-construction produces a simplicial set. The arrow category A r ( C ) {\displaystyle Ar(C)} of a category
K-theory_of_a_category
Textbook in topology
topology discussed as part of this presentation include simplicial complexes, fundamental groups, simplicial homology and singular homology, and the Poincaré
A Guide to the Classification Theorem for Compact Surfaces
A_Guide_to_the_Classification_Theorem_for_Compact_Surfaces
Theoretical object in mathematics
abstract simplicial complexes. One of the assumptions is a non-triviality condition: If the building is an n‑dimensional abstract simplicial complex,
Field_with_one_element
In mathematics, the curve complex is a simplicial complex C(S) associated to a finite-type surface S, which encodes the combinatorics of simple closed
Curve_complex
(F)} also have to be expressed as fractions (of the (n-1)-sphere). When the polytope is simplicial additional angle restrictions known as Perles relations
Gram–Euler_theorem
Mapping theorem in topology
neighborhood retracts are homotopy equivalent to finite simplicial complexes. First, by applying the simplicial approximation theorem, one shows that if f {\displaystyle
Lefschetz_fixed-point_theorem
Concept in mathematics
characterises hyperbolic spaces. Generalising the construction of the ends of a simplicial tree there is a natural notion of boundary at infinity for hyperbolic
Hyperbolic_metric_space
Concept in algebraic topology
being built on fairly concrete constructions (see also the related theory simplicial homology). In brief, singular homology is constructed by taking maps of
Singular_homology
Four-dimensional analog of the octahedron
fourth dimension)." Tyrrell & Semple 1971. Coxeter 1973, p. 130, § 7.6; "simplicial subdivision". Coxeter 1973, pp. 292–293, Table I(ii); "16-cell, 𝛽4".
16-cell
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SIMPLICIAL SPHERE
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SIMPLICIAL SPHERE
SIMPLICIAL SPHERE
SIMPLICIAL SPHERE
SIMPLICIAL SPHERE
SIMPLICIAL SPHERE
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