Fibrations with noncommutative fibers

TL;DR

This paper explores fibrations with noncommutative fibers in C*-algebra bundles, developing a spectral sequence to compute K-theory and demonstrating their natural occurrence through numerous examples.

Contribution

It introduces an analogue of the homotopy lifting property for noncommutative fibrations and derives a spectral sequence for K-theory computations.

Findings

Fibrations with noncommutative fibers are common in nature.

A spectral sequence for K-theory analogous to Leray-Serre is established.

Numerous examples illustrate the prevalence of such fibrations.

Abstract

We study an analogue of fibrations of topological spaces with the homotopy lifting property in the setting of C*-algebra bundles. We then derive an analogue of the Leray-Serre spectral sequence to compute the K-theory of the fibration in terms of the cohomology of the base and the K-theory of the fibres. We present many examples which show that fibrations with noncommutative fibres appear in abundance in nature.