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 $\left({E}_{p},{D}_{p}\right)$ an elliptic complex, its analytical index is the alternating sum

${\mathrm{ind}}_{\mathrm{an}}\left({E}_{p},{D}_{p}\right)=\sum _{p}\left(-1{\right)}^{+}\mathrm{dim}\mathrm{ker}{\Delta }_{p}\phantom{\rule{thinmathspace}{0ex}}.$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 January 24, 2013 19:40:10 by Urs Schreiber (82.113.99.233)