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Cohomology Operations

Theory of cohomology operations was developed by Steenrod for the singular cohomology theory .

There are several ways to construct cohomology operations in the singular cohomology theory. There is a purely algebraic treatment. See [ Woo97 ] by Wood and [ Smi ] by Larry Smith. It seems their q -deformations are defined [ HT04 ] and studied from algebraic viewpoints [ BGW ] .

We can also develope theory of cohomology operations in generalized cohomology theories , although they are much more complicated in general.

We also have notions of secondary and even higher cohomology operatioins, which can be used to detect topological informations that are invisible by primary operations. The Adams spectral sequence can be regarded as a systematic way of handling higher order cohomology operations.

Cohomology operations are also important for cohomology theories defined on categories other than the category of spaces. For example, Batanin, Berger, and Markl studied operads acting on the Hochschild cochains in [ BBM ] .

References