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An operation in spectral sequences that facilitates the computation of cohomology groups in fibration and allows detection of extension problems in cohomology.

A duality theorem between the cohomology of a space and the homology of its complement, providing a powerful tool in the study of manifolds and their embeddings.

A bilinear operation on cohomology groups, usually denoted by $\smile$, turning the cohomology ring into a graded algebra, which provides insight into the intersection patterns of subspaces in algebraic topology.

A product that pairs a cohomology class with a homology class to produce a homology class of lower dimension, providing a bridge between homology and cohomology theories.

A product operation on the homology groups of a space with a loop space structure that provides insight into loop structures and their geometric properties.

Associated with a short exact sequence of coefficient groups, it provides a way to relate different cohomology groups and detect the torsion in cohomology.

A higher-order cohomology operation generalizing the cup product that can obstruct the formality of a space and indicate the complexity of its cohomology ring structure.