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A type of cohomology theory that assigns a cohomology group to each integer dimension, using singular simplices and cochains.

De Rham cohomology uses differential forms on manifolds and has important applications in differential geometry and mathematical physics.

A cohomology theory based on the notion of covering spaces and provides tools for computing sheaf cohomology.

Sheaf cohomology is a method of assigning to each sheaf of abelian groups on a topological space a sequence of abelian cohomology groups, revealing global properties from local conditions.

Group cohomology measures the extent to which a group $G$ acts non-trivially on $G$-modules and can be used to classify group extensions.

The cohomology ring is an algebraic structure that is the direct sum of the cohomology groups of a space, with a graded ring structure given by the cup product.

A cohomology theory similar to singular cohomology but uses locally-defined cochains, developed by J.W. Alexander and E.H. Spanier.

Equivariant cohomology is a generalization of ordinary cohomology that takes into account group actions; it is often used in the context of G-spaces.

Floer cohomology provides invariants of 3-manifolds and the symplectic topology of 4-manifolds, and is inspired by the study of solutions to certain partial differential equations.