| dc.description.abstract | Given a complete n-dimensional Riemannian manifold X admitting a suitable action by a group Γ, each Γ-equivariant elliptic differential operator D gives rise to a higher index class IndΓ(D). A central strength of this algebraic invariant lies in the ability to encode important “symmetries" of X, with its construction involving differential geometry, K-theory, functional analysis, and C∗-algebraic notions. When a primary invariant such as IndΓ(D) is trivial, even finer geometrical and topological information can be obtained through the analysis of naturally occurring secondary higher invariants. An intrinsic issue is the general difficulty in computation of such invariants; generally speaking, the efficacy of the tools and techniques applied in order to reduce this computability difficulty depend crucially on the structure of the group Γ.
The content of this thesis concerns higher index theory in the setting of complete closed spin manifolds M with finitely generated and virtually nilpotent fundamental groups. A common approach is to pair the given secondary higher invariant with cyclic cohomology classes associated to the group algebra CΓ. The immediate question which arises is determining when this pairing can be rigorously well-defined; one aspect to be addressed is purely algebraic topological in nature, while another main difficulty involves norm estimates in functional calculus and subtle convergence issues. The first contribution of the thesis is to show that with respect to a virtually nilpotent group Γ every cyclic cocycle class on CΓ has a representative of polynomial growth. This cohomological growth condition is essential to proving that every cyclic cocycle class extends continuously from CΓ to certain geometric C∗-algebras, and provides the foundation for showing that under certain curvature assumptions and for π1(M) virtually nilpotent, the explicit integral formula describing a higher analogue of Lott’s delocalized eta invariant converges absolutely and is well-defined. We also use a determinant map construction of de la Harpe and Skandalis– adapted by Xie and Yu– to prove that if Γ is of polynomial growth then there is a well defined pairing between delocalized cyclic cocycles and K-theory classes of C∗-algebraic secondary higher invariants. When this K-theory class is that of a higher rho invariant of an invertible differential operator we show this pairing is precisely the aforementioned higher analogue of Lott’s delocalized eta invariant. As an application of this equivalence we provide a delocalized higher Atiyah-Patodi-Singer index theorem for compact spin manifolds with boundary, equipped with a positive scalar metric. | |
