On the Postnikov towers for real and complex connective K-theory

TL;DR

This paper explores the structure of Postnikov towers in real and complex connective K-theory, revealing how cofiber sequences related to complexification and Bott periodicity organize these towers.

Contribution

It demonstrates how cofiber sequences in real and complex K-theory factor through their Postnikov towers, uncovering a repeating pattern under Bott periodicity.

Findings

Identification of five distinct cofiber sequences in the towers.

Repetition of these sequences under Bott periodicity.

Deeper understanding of the relationship between real and complex K-theory.

Abstract

Complexification, from real connective K-theory to complex connective K-theory, sits in a well-known cofiber sequence between multiplication by eta and a map related to realification. We show how this cofiber sequence factors as one goes up the Postnikov towers of ko and ku. There are 5 distinct cofiber sequences which then repeat under Bott periodicity.