# nLab analytical index

### Context

#### Index theory

index theory, KK-theory

noncommutative stable homotopy theory

partition function

genus, orientation in generalized cohomology

## Definitions

operator K-theory

K-homology

integration

# Contents

## Idea

By pseudo-differential analysis? an elliptic operator acting on sections of two vector bundles on a manifold is a Fredholm operator and hence has closed kernel and cokernel of finite dimension. The difference of these two dimensions is the analytical index of the operator.

More generally, for $(E_p, D_p)$ an elliptic complex, its analytical index is the alternating sum

$ind_{an}(E_p, D_p) = \sum_p (-1)^+ dim (ker (\Delta_p)) \,.$

## Properties

This index does not the depend of the Sobolev space used to get a bounded operator (by elliptic regularity the kernel is made up of smooth sections and the same for the cokernel as it is the kernel of the adjoint). By topological K-theory one can associate to it also a topological index. The Atiyah-Singer index theorem say that this two indexes coincide.

Revised on March 30, 2014 08:59:31 by Urs Schreiber (89.204.154.204)