The Atiyah-Singer Index Formula for Subelliptic Operators on Contact Manifolds
Open Access
Author:
Van Erp, Johannes
Graduate Program:
Mathematics
Degree:
Doctor of Philosophy
Document Type:
Dissertation
Date of Defense:
August 15, 2004
Committee Members:
Nigel David Higson, Committee Chair/Co-Chair John Roe, Committee Member Victor Nistor, Committee Member Anna L Mazzucato, Committee Member Murat Gunaydin, Committee Member
Keywords:
Index theory Hypoelliptic operators Contact manifolds
Abstract:
The Atiyah-Singer index theorem gives a topological formula for the index of an elliptic differential operator. The topological index depends on a cohomology class that is constructed from the principal symbol of the operator.
On contact manifolds, the naturally arising geometric operators are not elliptic, but subelliptic. A filtration on the algebra of differential operators that is adapted to these geometric structures, naturally leads to a symbolic calculus that is noncommutative, and a corresponding subelliptic theory can be developed.
For such subelliptic operators we construct a symbol class in the K-theory of a noncommutative $C^*$-algebra naturally associated to the algebra of symbols. There is a canonical map from this noncommutative K-theory to the ordinary cohomology of the manifold, which gives a class to which
the Atiyah-Singer formula can be applied. In this way we define the topological index of a subelliptic operator, and we prove that it is equal to its analytic index.
