40 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
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
Functorial properties of Putnam's homology theory for Smale spaces
We investigate functorial properties of Putnam's homology theory for Smale
spaces. Our analysis shows that the addition of a conjugacy condition is
necessary to ensure functoriality. Several examples are discussed that
elucidate the need for our additional hypotheses. Our second main result is a
natural generalization of Putnam's Pullback Lemma from shifts of finite type to
non-wandering Smale spaces.Comment: Updated to agree with published versio
Dynamical correspondences for Smale spaces
We initiate the study of correspondences for Smale spaces. Correspondences are shown to provide a notion of a generalized morphism between Smale spaces and are a special case of finite equivalences. Furthermore, for shifts of finite type, a correspondence is related to a matrix which intertwines the adjacency matrices of the shifts. This observation allows us to define an equivalence relation on all Smale spaces which reduces to shift equivalence for shifts of finite type. Several other notions of equivalence are introduced on both correspondences and Smale spaces; a hierarchy between these equivalences is established. Finally, we provide several methods for constructing correspondences and provide specific examples