Equivariant index theory and non-positively curved manifolds.
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.