# nLab assembly map

Contents

### Context

#### 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

The analytic assembly map is a natural morphism from $G$-equivariant topological K-theory to the operator K-theory of a corresponding crossed product C*-algebra.

More generally in equivariant KK-theory this is called the Kasparov descent map and is of the form

$KK^G(A,B) \to KK(G \ltimes A, G \ltimes B)$

where on the left we have $G$-equivariant KK-theory and on the right ordinary KK-theory of crossed product C*-algebras (which by the discussion there are models for the groupoid convolution algebras of $G$-action groupoids).

## Properties

The Baum-Connes conjecture states that under some conditions the analytic assembly map is in fact an isomorphism. The Novikov conjecture makes statements about it being an injection. The Green-Julg theorem states that under some (milder) conditions the Kasparov desent map is an isomorphism.

## References

The construction goes back to

• Gennady Kasparov, The index of invariant elliptic operators, K-theory, and Lie group representations. Dokl. Akad. Nauk. USSR, vol. 268, (1983), 533-537.

An introduction is in

A textbook account is in