nLab
cyclic cohomology

under construction – for the moment see the closely related entry on Hochschild cohomology for more background

Idea

Cyclic cohomology is the $S^1$-equivariant cohomology refinement of Hochschild cohomology: the cohomology $H_{S^1}(ℒX,A)$ of free loop space objects equivariant under the natural circle action on $ℒX.$

Hochschild cohomology on $X$ – the cohomology of $ℒX$ – realizes (analogs of) differential forms on $X$. The circle action on $ℒX$ induces a differential on these forms. The $S^1$-invariant part of the cohomology is therefore given by (the analog of) closed differential forms.

References

• Jacek Brodzki?, An introduction to K-theory and cyclic cohomology (arXiv:funct-an/9606001)

• J. Cuntz, D. Quillen, Operators on noncommutative differential forms and cyclic homology Geometry, topology and physics, 77–111 Conf. Proc. Lecture Notes Geom. Topology, IV, Internat. Press, Cambridge, MA 1995

Lecture notes on Hochschild cohomology with some pointers to cyclic aspects are in section 4 and later of

• Victor Ginzburg, Lectures on noncommutative geometry (arXiv:math/0506603)

A lucid and inovative treatement of cyclic homology is in

• D. Kaledin, Cyclic homology with coefficients, math.KT/0702068, to appear in Yu. Manin’s 70th anniversary volume.

• D. Kaledin, Homological methods in noncommuttaive geometry, lecture notes, Tokyo 2008, pdf

To have a cyclic cohomology for an algebra over an operad, the operad needs to be a cyclic operad?:

• E. Getlzer, M. M. Kapranov, Cyclic operads and cyclic cohomology, in Geometry, topology and physics (1995), pp. 167–201, pdf