Cohomology

Cohomology assigns algebraic invariants to spaces and mathematical structures by taking cocycles modulo coboundaries. Its contravariant behavior often turns maps into maps in the opposite direction.

Cochain complex: A sequence of algebraic groups or modules connected by maps whose consecutive compositions are zero. Cohomology is computed from the kernels and images of its successive differentials.
Singular cohomology: The cohomology theory built from cochains on continuous maps from simplices into a topological space. It is the standard construction for extracting cohomological invariants from topological spaces.
Euler characteristic: An integer invariant formed as an alternating sum of ranks of homology or cohomology groups. Finite-dimensional cohomology groups recover this compact summary of a space's topology.
Homology: An algebraic theory that assigns graded invariants to spaces using cycles modulo boundaries. Homology is covariant in spaces, opposite to the usual contravariance of cohomology.
Henri Poincaré: A French mathematician whose work established foundational ideas in topology and dynamical systems. His study of manifolds and duality helped create the problems cohomology later organized.
Cocycle: An element in the kernel of a differential in a cochain complex. Cocycles are the elements eligible to represent cohomology classes.
De Rham cohomology: The cohomology of the differential forms on a smooth manifold under the exterior derivative. It realizes cohomology through differential forms and integration.
Chern class: A sequence of cohomology classes that encodes characteristic information about a complex vector bundle. Cohomology provides the groups in which these bundle invariants live.
Homotopy group: A group recording homotopy classes of based maps from spheres into a space. Homotopy groups retain nonlinear information that cohomology can lose.
Élie Cartan: A French mathematician who developed differential forms and major parts of Lie theory. His exterior differential calculus became a central model for cohomological constructions.

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