644 research outputs found
Analytic and topological index maps with values in the K-theory of mapping cones
Index maps taking values in the -theory of a mapping cone are defined and
discussed. The resulting index theorem can be viewed in analogy with the
Freed-Melrose index theorem. The framework of geometric -homology is used in
a fundamental way. In particular, an explicit isomorphism from a geometric
model for -homology with coefficients in a mapping cone, , to
is constructed.Comment: 22 page
Relative geometric assembly and mapping cones, Part I: The geometric model and applications
Inspired by an analytic construction of Chang, Weinberger and Yu, we define
an assembly map in relative geometric -homology. The properties of the
geometric assembly map are studied using a variety of index theoretic tools
(e.g., the localized index and higher Atiyah-Patodi-Singer index theory). As an
application we obtain a vanishing result in the context of manifolds with
boundary and positive scalar curvature; this result is also inspired and
connected to work of Chang, Weinberger and Yu. Furthermore, we use results of
Wahl to show that rational injectivity of the relative assembly map implies
homotopy invariance of the relative higher signatures of a manifold with
boundary.Comment: 37 pages. Accepted in Journal of Topolog
Relative geometric assembly and mapping cones, Part II: Chern characters and the Novikov property
We study Chern characters and the assembly mapping for free actions using the
framework of geometric -homology. The focus is on the relative groups
associated with a group homomorphism along with
applications to Novikov type properties. In particular, we prove a relative
strong Novikov property for homomorphisms of hyperbolic groups and a relative
strong -Novikov property for polynomially bounded homomorphisms of
groups with polynomially bounded cohomology in \C. As a corollary, relative
higher signatures on a manifold with boundary , with belonging to the class above, are homotopy invariant.Comment: 32 pages, accepted in M\"unster Journal of Mathematic
Group actions on Smale space C*-algebras
Group actions on a Smale space and the actions induced on the C*-algebras
associated to such a dynamical system are studied. We show that an effective
action of a discrete group on a mixing Smale space produces a strongly outer
action on the homoclinic algebra. We then show that for irreducible Smale
spaces, the property of finite Rokhlin dimension passes from the induced action
on the homoclinic algbera to the induced actions on the stable and unstable
C*-algebras. In each of these cases, we discuss the preservation of
properties---such as finite nuclear dimension, Z-stability, and classification
by Elliott invariants---in the resulting crossed products.Comment: 30 pages. Final version, to appear in Ergodic Theory Dynam. System
Smale space C*-algebras have nonzero projections
The main result of the present paper is that the stable and unstable
C*-algebras associated to a mixing Smale space always contain nonzero
projections. This gives a positive answer to a question of the first listed
author and Karen Strung and has implications for the structure of these
algebras in light of the Elliott program for simple C*-algebras. Using our main
result, we also show that the homoclinic, stable, and unstable algebras each
have real rank zero.Comment: 15 page
The bordism group of unbounded KK-cycles
We consider Hilsum's notion of bordism as an equivalence relation on
unbounded -cycles and study the equivalence classes. Upon fixing two
-algebras, and a -subalgebra dense in the first -algebra, a
-graded abelian group is obtained; it maps to the
Kasparov -group of the two -algebras via the bounded transform. We
study properties of this map both in general and in specific examples. In
particular, it is an isomorphism if the first -algebra is the complex
numbers (i.e., for -theory) and is a split surjection if the first
-algebra is the continuous functions on a compact manifold with boundary
when one uses the Lipschitz functions as the dense -subalgebra.Comment: 38 page
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