Simplicial homologyIn 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 (the case of dimension 0). Simplicial homology arose as a way to study topological spaces whose building blocks are n-simplices, the n-dimensional analogs of triangles. This includes a point (0-simplex), a line segment (1-simplex), a triangle (2-simplex) and a tetrahedron (3-simplex).
Cellular homologyIn mathematics, cellular homology in algebraic topology is a homology theory for the category of CW-complexes. It agrees with singular homology, and can provide an effective means of computing homology modules. If is a CW-complex with n-skeleton , the cellular-homology modules are defined as the homology groups Hi of the cellular chain complex where is taken to be the empty set. The group is free abelian, with generators that can be identified with the -cells of . Let be an -cell of , and let be the attaching map.
Reduced homologyIn mathematics, reduced homology is a minor modification made to homology theory in algebraic topology, motivated by the intuition that all of the homology groups of a single point should be equal to zero. This modification allows more concise statements to be made (as in Alexander duality) and eliminates many exceptional cases (as in the homology groups of spheres). If P is a single-point space, then with the usual definitions the integral homology group H0(P) is isomorphic to (an infinite cyclic group), while for i ≥ 1 we have Hi(P) = {0}.
Disk (mathematics)In geometry, a disk (also spelled disc) is the region in a plane bounded by a circle. A disk is said to be closed if it contains the circle that constitutes its boundary, and open if it does not. For a radius, , an open disk is usually denoted as and a closed disk is . However in the field of topology the closed disk is usually denoted as while the open disk is . In Cartesian coordinates, the open disk of center and radius R is given by the formula: while the closed disk of the same center and radius is given by: The area of a closed or open disk of radius R is πR2 (see area of a disk).
Hurewicz theoremIn mathematics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz homomorphism. The theorem is named after Witold Hurewicz, and generalizes earlier results of Henri Poincaré. The Hurewicz theorems are a key link between homotopy groups and homology groups. For any path-connected space X and positive integer n there exists a group homomorphism called the Hurewicz homomorphism, from the n-th homotopy group to the n-th homology group (with integer coefficients).
Chain (algebraic topology)In algebraic topology, a -chain is a formal linear combination of the -cells in a cell complex. In simplicial complexes (respectively, cubical complexes), -chains are combinations of -simplices (respectively, -cubes), but not necessarily connected. Chains are used in homology; the elements of a homology group are equivalence classes of chains. For a simplicial complex , the group of -chains of is given by: where are singular -simplices of . Note that any element in not necessary to be a connected simplicial complex.
Relative homologyIn algebraic topology, a branch of mathematics, the (singular) homology of a topological space relative to a subspace is a construction in singular homology, for pairs of spaces. The relative homology is useful and important in several ways. Intuitively, it helps determine what part of an absolute homology group comes from which subspace. Given a subspace , one may form the short exact sequence where denotes the singular chains on the space X. The boundary map on descends to and therefore induces a boundary map on the quotient.
Morse theoryIn mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiable function on a manifold will reflect the topology quite directly. Morse theory allows one to find CW structures and handle decompositions on manifolds and to obtain substantial information about their homology.
Combinatorial topologyIn mathematics, combinatorial topology was an older name for algebraic topology, dating from the time when topological invariants of spaces (for example the Betti numbers) were regarded as derived from combinatorial decompositions of spaces, such as decomposition into simplicial complexes. After the proof of the simplicial approximation theorem this approach provided rigour. The change of name reflected the move to organise topological classes such as cycles-modulo-boundaries explicitly into abelian groups.
Abstract simplicial complexIn 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 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 (sets of size 3), their edges (sets of size 2), and their vertices (sets of size 1).
