nLab analytical index



Index theory

Integration theory



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) pdim(ker(D p)). ind_{an}(E_p, D_p) = \sum_p (-1)^p dim (ker (D_p)) \,.


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.

Last revised on November 6, 2016 at 00:40:35. See the history of this page for a list of all contributions to it.