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Title page for ETD etd-04022007-140838


Type of Document Dissertation
Author Shan, Lin
Author's Email Address lin.shan@vanderbilt.edu
URN etd-04022007-140838
Title Equivariant index theory and non-positively curved manifolds.
Degree PhD
Department Mathematics
Advisory Committee
Advisor Name Title
Guoliang Yu Committee Chair
Bruce Hughes Committee Member
Dietmar Bisch Committee Member
Gennadi Kasparov Committee Member
Thomas Kephart Committee Member
Keywords
  • higher index
  • non-positively curved manifolds
  • Index theory
  • Novikov Conjecture.
Date of Defense 2007-03-29
Availability unrestricted
Abstract
An elliptic differential operator D on a compact manifold M is a Fredholm operator. The only

topological invariant for a Fredholm operator is the Fredholm index [Dou72], which is defined to

be dim(kerD) − dim(cokerD). Fredholm index is a homotopy invariant. The Atiyah-Singer index

theorem calculates the Fredholm index of D in terms of its symbol sigma(D) and M. This theorem

establishes a bridge between analysis, geometry and topology [AS1, AS3]. Index theorems have been generalized to noncompact manifolds of various sorts. Elliptic operators on noncompact manifolds are no longer Fredholm in the classical sense, but are Fredholm in a

generalized sense with respect to certain operator algebras. An important topological invariant for an elliptic operator is the generalized Fredholm index, which lives in the K-theory of an operator algebra.

In this thesis we define the equivariant index map for proper group actions and prove that this

equivariant index map is injective for certain manifolds and groups. We also prove that the index map [Y95, Y97] is injective for spaces which admit a coarse embedding into a simply-connected complete Riemannian manifold with nonpositive sectional curvature, which is the joint work with Qin Wang.

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