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...

probably the simplest homology theory to use is graph homology, which could be regarded as a 1-dimensional special case of simplicial homology, the latter of...

the choice of a homeomorphism in a suitable simplicial complex. Spaces being homeomorphic to a simplicial complex are called triangulable. Triangulation...

concrete constructions (see also the related theory simplicial homology). In brief, singular homology is constructed by taking maps of the standard n-simplex...

some statements in homology-theory one wishes to replace simplicial complexes by a subdivision. On the level of simplicial homology groups one requires...

of parameters. To find the persistent homology of a space, the space must first be represented as a simplicial complex. A distance function on the underlying...

simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a simplicial complex is an abstract simplicial...

to Emmy Noether. Betti numbers are used today in fields such as simplicial homology, computer science and digital images. Informally, the kth Betti number...

boundary) of dimension at least 5 are homeomorphic to simplicial complexes if and only if there is a homology 3 sphere Σ with Rokhlin invariant 1 such that the...

"holes" in the graph. It is a special case of a simplicial homology, as a graph is a special case of a simplicial complex. Since a finite graph is a 1-complex...

In mathematics, the simplicial approximation theorem is a foundational result for algebraic topology, guaranteeing that continuous mappings can be (by...

In mathematics, a simplicial set is an object composed of simplices in a specific way. Simplicial sets are higher-dimensional generalizations of directed...

Mayer–Vietoris sequence holds for a variety of cohomology and homology theories, including simplicial homology and singular cohomology. In general, the sequence holds...

work with. The fundamental group of a (finite) simplicial complex does have a finite presentation. Homology and cohomology groups, on the other hand, are...

connected. Chains are used in homology; the elements of a homology group are equivalence classes of chains. For a simplicial complex X {\displaystyle X}...

completely determine its homology groups with coefficients in A, for any abelian group A: Hi(X; A) Here Hi might be the simplicial homology, or more generally...

an optimization method with inequality constraints Simplicial complex Simplicial homology Simplicial set Spectrahedron Ternary plot Elte, E.L. (2006) [1912]...

A simplicial map (also called simplicial mapping) is a function between two simplicial complexes, with the property that the images of the vertices of...

simplicial norm measures the complexity of homology classes. Given a closed and oriented manifold, one defines the simplicial norm by minimizing the sum of the...

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...

spaces often admit a model category structure, such as the category of simplicial sets. Another model category is the category of chain complexes of R-modules...

In combinatorics, an abstract simplicial complex (ASC), often called an abstract complex or just a complex, is a family of sets that is closed under taking...

independence complex of an undirected graph G, denoted by I(G), is an abstract simplicial complex (that is, a family of finite sets closed under the operation of...

Cambridge University Press, ISBN 0-521-79540-0. Detailed discussion of homology theories for simplicial complexes and manifolds, singular homology, etc....

In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated...

Poincaré's papers introduced the concepts of the fundamental group and simplicial homology, provided an early formulation of the Poincaré duality theorem, introduced...

chain homotopy corresponds to a simplicial homotopy) Simplicial homology Goerss, Paul G.; Jardin, John F. (2009). Simplicial Homotopy Theory. Birkhäuser Basel...

morphism corresponding to the homology functor. We build the pushforward homomorphism as follows (for singular or simplicial homology): First, the map f : X...

Collapses find applications in computational homology. Let K {\displaystyle K} be an abstract simplicial complex. Suppose that τ , σ {\displaystyle \tau...

commutative algebra. A simplicial complex Δ is Cohen–Macaulay over k if and only if for all simplices σ ∈ Δ, all reduced simplicial homology groups of the link...