Abstract
There are several ways of modifying cyclic homology: by altering the cyclic bicomplex, by putting up other groups than the cyclic groups or by enlarging the category of algebras.
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Bibliographical Comments on Chapter 5
The idea of looking at the periodic complex, and so at the periodic theory, is already in the seminal article of Connes [C], see also Goodwillie [1985a]. The idea that the theory HC- is relevant is due to Hood-Jones [1987], where they recognize this theory as the dual of the cyclic theory over Ha. (k). Similar statements can be found in Feigin-Tsygan [FT]. The product structure on HC- was introduced in Hood-Jones [1987] by using the acyclic model technique. De Rham cohomology has been generalized to crystalline cohomology by Grothendieck and the comparison with the periodic cyclic theory is done in Feigin-Tsygan [1987], see also Kassel [1987, cor. 3.12]. In the literature periodic cyclic homology is denoted either by HCP (adopted here), or PHC,or HCP,or HP,or even simply H.
Dihedral and quaternionic homology were introduced and studied in Loday [1987]. Independent and similar work appeared in Krasauskas-Lapin-Solovev [1987] and Krasauskas-Solovev [1986, 1988 ]. Subsequent work was done in Lodder [1990, 1992] and in Dunn [1989] where the relationship with 0(2)-spaces is also worked out. An interesting application to higher Arf invariants is done in Wolters [1992].
3-4. The extension of HC to DG-algebras appeared in Vigué-Burghelea [1985] and also Goodwillie [1985a]. The idea of getting a decomposition of HC from this point of view is in Burghelea-Vigué [1988]. Extensive computations have been made in loc. cit., Brylinski [1987b], Vigué [1988, 1990), Geller-Reid-Weibel [1989], Bach [1992], Hanlon [ 1986 ]. Some of these results can be found in Feigin-Tsygan [FT].
Bivariant cyclic cohomology was taken out from Jones-Kassel [1989], see also Kassel [1989a]. The A-decomposition is in Nuss [1992].
Some computations in the topological framework are done in Connes [C]. En-tire cyclic cohomology is treated in Connes [1988] and used extensively for the proof of some cases of the Novikov conjecture in Connes-Moscovici [1990] and Connes-Gromov-Moscovici [19901). Further work can be found in Connes-Gromov-Moscovici [1992]) (asymptotic cyclic cohomology, again in relationship with the Novikov con-jecture). Many other papers relating the index theory and the entire cyclic coho-mology are listed in the references.
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© 1998 Springer-Verlag Berlin Heidelberg
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Loday, JL. (1998). Variations on Cyclic Homology. In: Cyclic Homology. Grundlehren der mathematischen Wissenschaften, vol 301. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-11389-9_5
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DOI: https://doi.org/10.1007/978-3-662-11389-9_5
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