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

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

Etale cohomology is a tool in algebraic geometry for studying the properties of schemes, particularly over fields other than the complex numbers, by means of sheaves defined with respect to the étale topology.

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.

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

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

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

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