nLab
analytical index

Context

Index theory

Integration theory

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)(E_p, D_p) an elliptic complex, its analytical index is the alternating sum

ind an(E p,D p)= p(1) +dim(ker(Δ p)). 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)