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Advances in Noncommutative Geometry pp 291–319Cite as

Cyclic homology in a special world

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Abstract

In work of Connes and Consani, Γ-spaces have taken a new importance. Segal introduced Γ-spaces in order to study stable homotopy theory, but the new perspective makes it apparent that also information about the unstable structure should be retained. Hence, the question naturally presents itself: to what extent are the commonly used invariants available in this context? We offer a quick survey of (topological) cyclic homology and point out that the categorical construction is applicable also in an \({\mathbb N}\)-algebra (aka. semi-ring or rig) setup.

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Acknowledgements

Apart from obvious input from Alain Connes this tribute has benefitted from enlightening conversations with M. Brun, K. Consani, L. Hesselholt, M. Hill, and C. Schlichtkrull. Also, the preprint [8] by de Brito and Moerdijk and the papers of Santhanam [55] and Mandell [46] were inspirational. The author also wants to thank an anonymous referee for correcting an unfortunate misconception about events in the early 1980s.

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Authors and Affiliations

  1. University of Bergen, Bergen, Norway

    Bjørn Ian Dundas

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  1. Bjørn Ian Dundas
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Correspondence to Bjørn Ian Dundas .

Editor information

Editors and Affiliations

  1. Department of Physics Institute for Mathematics, Astrophysics and Particle Physics, Nijmegen, The Netherlands

    Ali Chamseddine

  2. Department of Mathematics, Johns Hopkins University, Baltimore, MD, USA

    Caterina Consani

  3. Department of Mathematics, Pennsylvania State University, University Park, PA, USA

    Nigel Higson

  4. Department of Mathematics, The University of Western Ontario, London, ON, Canada

    Masoud Khalkhali

  5. Department of Mathematics, Ohio State University, Columbus, OH, USA

    Henri Moscovici

  6. Department of Mathematics, Texas A&M University, College Station, TX, USA

    Guoliang Yu

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Dundas, B.I. (2019). Cyclic homology in a special world. In: Chamseddine, A., Consani, C., Higson, N., Khalkhali, M., Moscovici, H., Yu, G. (eds) Advances in Noncommutative Geometry. Springer, Cham. https://doi.org/10.1007/978-3-030-29597-4_5

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