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 AbstractAn elliptic differential operator D on a compact manifold M is a Fredholm operator. The onlytopological 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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