Operations in connective K-theory

TL;DR

This paper classifies additive and multiplicative operations in connective K-theory with various coefficients, revealing the structure of stable operations and their generation by Adams operations, especially highlighting differences between integral and torsion-free coefficients.

Contribution

It provides a comprehensive classification of additive and multiplicative operations in connective K-theory, including the topological basis and the role of Adams operations, with new insights into integral versus torsion-free cases.

Findings

Integral additive operations involve $\\hat{\mathbb{Z}}$-coefficients.

Stable multiplicative operations generate homogeneous additive stable operations.

Integral operations are not solely generated by Adams operations.

Abstract

In this article 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 $\hat{{\mathbb{Z}}}$ one. 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 $\hat{{\mathbb{Z}}}$-coefficients are topologically generated by stable multiplicative operations. This is not true for integral operations.