Power operations and coactions in highly commutative homology theories

Baker, A. (2015) Power operations and coactions in highly commutative homology theories. Publications of the Research Institute for Mathematical Sciences, 51(2), pp. 237-272. (doi: 10.4171/PRIMS/154)

Abstract

Power operations in the homology of infinite loop spaces, and H∞ or E∞ ring spectra have a long history in Algebraic Topology. In the case of ordinary mod p homology for a prime p, Dyer-Lashof operations interact with Steenrod operations via the Nishida relations, but for many purposes this leads to complicated calculations once iterated applications of these functions are required. On the other hand, the homology coaction turns out to provide tractable formulae better suited to exploiting multiplicative structure. We show how to derive suitable formulae for the interaction between power operations and homology coactions in a wide class of examples; our approach makes crucial use of modern frameworks for spectra with well behaved smash products. In the case of mod <i>p</i> homology, our formulae extend those of Bisson & Joyal to odd primes. We also show how to exploit our results in sample calculations, and produce some apparently new formulae for the Dyer-Lashof action on the dual Steenrod algebra.

• Item Type: Articles (Other)
• Keywords: Power operations; coactions; E∞ and H∞ ring spectra
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Baker, Dr Andrew
• Authors: Baker, A.
• Subjects: Q Science > QA Mathematics
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Research Group: Geometry & Topology
• Journal Name: Publications of the Research Institute for Mathematical Sciences
• Publisher: EMS Publishing House
• ISSN: 0034-5318
• ISSN (Online): 1663-4926
• Copyright Holders: Copyright © 2015 Research Institute for Mathematical Sciences, Kyoto University.
• First Published: First published in Publications of the Research Institute for Mathematical Sciences 51(2):237-272
• Publisher Policy: Reproduced with the permission of the publisher.

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