# nLab noncommutative stable homotopy theory

### Context

#### Noncommutative geometry

noncommutative geometry

(geometry $\leftarrow$ Isbell duality $\to$ algebra)

## Relation to physics

#### Stable Homotopy theory

stable homotopy theory

Introduction

# Contents

#### Index theory

index theory, KK-theory

noncommutative stable homotopy theory

partition function

genus, orientation in generalized cohomology

## Definitions

operator K-theory

K-homology

# Contents

## Idea

By noncommutative stable homotopy theory is meant an analog of classical stable homotopy theory obtained as one passes from topology to noncommutative topology, hence as one generalizes topological spaces to non-neccesarily commutative (formal duals of) C*-algebras.

The concrete implementation of such noncommutative stable homotopy theory is usually taken to be KK-theory or rather E-theory. See there for details.

## References

A reference that mentions the term “noncommutative stable homotopy theory” explicitly is

But see at KK-theory and E-theory for more references.

Last revised on May 20, 2013 at 12:43:04. See the history of this page for a list of all contributions to it.