Operations in connective K-theory

Merkurjev, Alexander; Vishik, Alexander

doi:10.2140/ant.2023.17.1595

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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 $\hat{ℤ}$ 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 $\hat{ℤ}$ -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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Vol. 17, No. 9, 2023