Vol. 6, No. 3, 2021
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The plethory of operations in complex topological $K\mkern-2mu$-theory

William Mycroft and Sarah Whitehouse
Vol. 6 (2021), No. 3, 481–502
DOI: 10.2140/akt.2021.6.481
Abstract

We provide a concrete introduction to the topologised, graded analogue of an algebraic structure known as a plethory, originally due to Tall and Wraith. Stacey and Whitehouse showed this structure is present on the cohomology operations for a suitable generalised cohomology theory. We compute an explicit expression for the plethory of operations for complex topological K-theory. This is formulated in terms of a plethory enhanced with structure corresponding to the looping of operations. In this context we show that the familiar λ-operations generate all the operations.

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Keywords
topological $K\mkern-2mu$-theory, $K\mkern-2mu$-theory operations, lambda operations, lambda ring, plethory
Mathematical Subject Classification 2010
Primary: 55N15
Secondary: 55S25
Milestones
Received: 15 January 2020
Revised: 26 January 2021
Accepted: 20 February 2021
Published: 11 September 2021
Authors
William Mycroft
London
United Kingdom
Sarah Whitehouse
School of Mathematics and Statistics
University of Sheffield
Sheffield
United Kingdom