1,757 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
Realizing the analytic surgery group of Higson and Roe geometrically, Part II: Relative eta-invariants
We apply the geometric analog of the analytic surgery group of Higson and Roe
to the relative -invariant. In particular, by solving a Baum-Douglas type
index problem, we give a "geometric" proof of a result of Keswani regarding the
homotopy invariance of relative -invariants. The starting point for this
work is our previous constructions in "Realizing the analytic surgery group of
Higson and Roe geometrically, Part I: The geometric model" (arXiv:1308.5990).Comment: 38 pages, to appear in Mathematische Annale
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
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
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