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Operations in connective K-theory

Alexander Merkurjev and Alexander Vishik

Vol. 17 (2023), No. 9, 1595–1636
DOI: 10.2140/ant.2023.17.1595
Abstract

We classify additive operations in connective K-theory with various torsion-free coefficients. We discover that the answer for the integral case requires understanding of the ^ case. Moreover, although integral additive operations are topologically generated by Adams operations, these are not reduced to infinite linear combinations of the latter ones. We describe a topological basis for stable operations and relate it to a basis of stable operations in graded K-theory. We classify multiplicative operations in both theories and show that homogeneous additive stable operations with ^-coefficients are topologically generated by stable multiplicative operations. This is not true for integral operations.

Keywords
connective K-theory, additive operations, oriented cohomology theories
Mathematical Subject Classification
Primary: 19L20, 19L41, 55S25
Milestones
Received: 26 January 2022
Revised: 17 August 2022
Accepted: 4 October 2022
Published: 9 September 2023
Authors
Alexander Merkurjev
Department of Mathematics
University of California
Los Angeles, CA
United States
Alexander Vishik
School of Mathematical Sciences
University of Nottingham
Nottingham
United Kingdom

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