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...

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...

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

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

mathematics, triangulation describes the replacement of topological spaces with simplicial complexes by the choice of an appropriate homeomorphism. A space that...

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

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

Here H i {\displaystyle H_{i}} might be the simplicial homology, or more generally the singular homology. The usual proof of this result is a pure piece...

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

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

"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...

dimensions of a simplicial sphere. In December 2018, the g-conjecture was proven by Karim Adiprasito in the more general context of rational homology spheres...

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

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...

algebraic topology, simplicial complexes are often useful for concrete calculations. For the definition of homology groups of a simplicial complex, one can...

mathematics, a simplicial set is a sequence of sets with internal order structure (abstract simplices) and maps between them. Simplicial sets are higher-dimensional...

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...

then simplicial intersection homology groups can be defined in a similar way, and are naturally isomorphic to the singular intersection homology groups...

a manifold. simplicial homology A simplicial homology is the (canonical) homology of a simplicial complex. Note it applies to simplicial complexes and...

1\otimes a_{i+1}\otimes \cdots \otimes a_{n}.} Hochschild homology is the homology of this simplicial module. There is a similar looking complex B ( A / k...

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...

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

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

set Simplicial category Chain (algebraic topology) Betti number Euler characteristic Genus Riemann–Hurwitz formula Singular homology Cellular homology Relative...

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

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

contractible) then the Lefschetz number is one because the only non-zero simplicial homology group is: H 0 ( B ) {\displaystyle H_{0}(B)} and f acts as the identity...

complex may not be. Čech cohomology Computational geometry Simplicial complex Simplicial homology Ghrist, Robert W. (2014). Elementary applied topology (1st ed...

corresponds to a simplicial homotopy) Simplicial homology Homotopy category of an ∞-category Goerss, Paul G.; Jardin, John F. (2009). Simplicial Homotopy Theory...