Homology (mathematics)

Definition

In mathematics, the term homology, originally introduced in algebraic topology, has three primary, closely related usages. First, there is the homology of a chain complex, a sequence of abelian groups, called homology groups, which are regarded fundamental invariants of the chain complex. Secondly, when one can associate a chain complex to a different mathematical object, one can also associate its homology to that object. Distinct procedures of associating chain complexes to a given object are grouped into homology theories. Finally, homology is important in the study of topological spaces. Under nice conditions in which distinct homology theories for a single topological space produce the same homology groups, one can define a single homology of a topological space. This last notion of homology is closely related to topological ideas frequently discussed in popular mathematics such as the holes of a surface or the cycles of a graph. There is also a related notion of the cohomology of a cochain complex, giving rise to various cohomology theories, in addition to the notion of the cohomology of a topological space.

Related concepts

Abelian category Abelian group Abelianization Abstract algebra Algebraic topology Algebraic variety American Mathematical Society Analysis Situs (paper)Antipodal point Ball (mathematics)Banach fixed-point theorem Bar-Ilan University Bernhard Riemann Betti number Borel–Moore homology Borsuk–Ulam theorem Boundary-value problem Boundary map Boy surface Braid theory Brouwer fixed point theorem Bundle (mathematics)C++CW complex Category (mathematics)Category theory Cellular homology Chain (algebraic topology)Chain complex Chern class Circle Closed set Closure (topology)Cobordism Cochain complex Cochain complexes Cohomology Cohomology class Combinatorial topology Compact space Computational electromagnetics Cone (topology)Connected space Connected sum Continuity (topology)Continuous function Continuous function (topology)Continuous map Continuum (topology)Contractible topological space Cross-cap Cycle (graph theory)Cycle space Cyclic group Cyclic homology David Richeson De Rham cohomology Derived category Derived functor Differential topology Digital topology Direct product of groups Discrete Morse theory Dynamical system Edwin Spanier Eilenberg–Steenrod axioms Embedding Emmy Noether Enrico Betti Equivalence class Euclidean 3-space Euclidean space Euler characteristic Exact functor Exact sequence Finite element methods Finitely-generated abelian group Floer homology Formal sum Free abelian group Free group Free module Friedrich Hirzebruch Functor Fundamental group General topology Generator (groups)Genus (mathematics)Genus (topology)Genus g surface Geometric topology Gmsh Graph homology Great circle Group (mathematics)Group cohomology Group extension Group homomorphism Hairy ball theorem Hamel dimension Hausdorff space Henri Cartan Henri Poincaré Hochschild homology Hodge-Laplace operator Hole (topology)Homeomorphism Homological algebra Homological conjectures in commutative algebra Homological connectivity Homological dimension Homology (disambiguation)Homotopy Homotopy group Homotopy groups of spheres Homotopy theory Hurewicz theorem Image (mathematics)Injective Interior (topology)Intersection homology Invariance of domain Invariant manifold John Stillwell Jordan curve theorem June Barrow-Green K-homology KAM theorem Kernel (algebra)Kernel (category theory)Khovanov homology Klein bottle Knot theory Künneth theorem Leopold Vietoris List of algebraic topology topics List of cohomology theories List of general topology topics List of geometric topology topics List of important publications in mathematics List of topology topics Long exact sequence Low-dimensional topology Manifold Map (mathematics)Mathematics Matrix (mathematics)Mayer-Vietoris sequence Metric space Mina Teicher Model category Module (mathematics)Module homomorphism Morse homology Morse theory Möbius strip N-sphere Network topology Normal subgroup Open set Orientability Oxford University Press Periodic orbit Persistent homology Peter Hilton Physics Poincaré conjecture Poincaré duality Point cloud Pointless topology Polyhedral complex Popular mathematics Presentation of a group Product topology Projective plane Quotient group Quotient module Rank of an abelian group Reduced homology Relative homology Resolution (algebra)Ring (mathematics)Rose (topology)Samuel Eilenberg Second-countable space Sensor network Set-theoretic topology Short exact sequence Simple-homotopy equivalence Simplex Simplicial complex Simplicial homology Singular homology Smith normal form Smooth manifold Sphere Steenrod homology Surjective The Princeton Companion to Mathematics Timothy Gowers Topological data analysis Topological manifold Topological property Topological space Topology Tor functor Torsion (algebra)Torus Trivial group Tychonoff's theorem Uniform space Universal coefficient theorem Urysohn's lemma Vertex (geometry)Walther Mayer Winding number Zig-zag lemma Čech cohomology

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