group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
The graded subspace of Hochschild complex of an qassociative algebra consisting of invariant chains under certain actions of symmetric groups, is in fact a subcomplex, the cyclic subcomplex. Cyclic homology is Hochschild homology on cyclically invariant chains.
There are many versions (e.g. periodic cyclic homology) and generalities of the notion (e.g. for schemes, for algebras over cyclic operads). Topological cyclic homology? is an adaptation to ring spectra.
We may understand Hochschild homology as the cohomology of free loop space objects (as described there). These free loop space objects are canonically equipped with a circle group-action that rotates the loops. Cyclic homology is the corresponding $S^1$-equivariant cohomology of free loop space objects.
Let $A$ be an associative algebra over a ring $k$. Write $C_\bullet(A,A)$ for the Hochschild homology chain complex of $A$ with coefficients in $A$.
For each $n \in \mathbb{N}$ let $\lambda : C_n(A,A) \to C_n(A,A)$ be the $k$-linear map that cyclically permutes the elements and introduces a sign:
The cyclic homology complex $C^\lambda_\bullet(A)$ of $A$ is the quotient of the Hochschild homology complex of $A$ by cyclic permutations:
The homology of the cyclic complex, denoted
is called the cyclic homology of $A$.
If $I\subset A$ is an ideal, then the relative cyclic homology groups $HC_n(A,I)$ are the homology groups of the complex $C_\bullet(A,I) = ker(C_\bullet(A)\to C_\bullet(A/I))$.
The complex for cyclic homology was considered by Alain Connes and Boris Tsygan around 1981, 1982.
Monographs:
Quick lecture notes:
D. Kaledin, Tokyo lectures “Homological methods in non-commutative geometry”, (pdf, TeX); and related but different Seoul lectures
Masoud Khalkhali, A short survey of cyclic cohomology, arxiv/1008.1212
Some influential original references from 1980s:
Some modern illuminating references: