Homology group

A homology group records cycles in a space modulo cycles that bound higher-dimensional chains. Its elements distinguish holes that are invisible to lower-dimensional boundaries.

Chain complex: A sequence of abelian groups or modules connected by boundary maps whose successive compositions are zero. Homology groups are computed from the kernels and images of consecutive maps in a chain complex.
Topological space: A set equipped with a notion of open sets that specifies its topological structure. Homology assigns algebraic invariants to spaces while remaining insensitive to continuous deformation.
Betti number: The rank of a homology group, counting independent classes in a given dimension. Betti numbers summarize the free part of each homology group as counts of holes.
Cohomology group: A group built from cocycles modulo coboundaries, dual in construction to homology. Cohomology uses cochains and often carries additional products, unlike homology's cycle classes.
Boundary operator: A homomorphism sending each chain to its boundary, with the boundary of a boundary equal to zero. Its kernel gives cycles, while its image gives boundaries used in the homology quotient.
Singular simplex: A continuous map from a standard simplex into a topological space. Singular simplices provide chains that define singular homology for general spaces.
Euler characteristic: An alternating sum of topological counts that is invariant under homotopy equivalence for suitable spaces. It can be calculated as an alternating sum of the ranks of homology groups.
Homotopy group: A group of based homotopy classes of maps from spheres into a pointed space. Homotopy groups retain nonlinear information that homology can discard.
Cycle (homology): A chain whose boundary is zero. Cycles are the elements retained before boundaries are identified with zero.
Singular homology: A homology theory built from formal sums of continuous maps of simplices into a space. It is a standard construction yielding homology groups for broad classes of spaces.

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