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Register a new userThe adiabatic groupoid and the Higson-Roe exact sequence
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Authors
Vito Felice Zenobi
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Let $widetilde{X}$ be a smooth Riemannian manifold equipped with a proper, free, isometric and cocompact action of a discrete group $Gamma$. In this paper we prove that the analytic surgery exact sequence of Higson-Roe for $widetilde{X}$ is isomorphic to the exact sequence associated to the adiabatic deformation of the Lie groupoid $widetilde{X}times_Gammawidetilde{X}$. We then generalize this result to the context of smoothly stratified manifolds. Finally, we show, by means of the aforementioned isomorphism, that the $varrho$-classes associated to a metric with positive scalar curvature defined by Piazza and Schick corresponds to the $varrho$-classes defined by the author of this paper.
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The main result of this paper is a new and direct proof of the natural transformation from the surgery exact sequence in topology to the analytic K-theory sequence of Higson and Roe. Our approach makes crucial use of analytic properties and new index theorems for the signature operator on Galois coverings with boundary. These are of independent interest and form the second main theme of the paper. The main technical novelty is the use of large scale index theory for Dirac type operators that are perturbed by lower order operators.
In this article we extend the Bloch-Wigner exact sequence over local rings, where their residue fields have more than nine elements. Moreover, we prove Van der Kallens theorem on the presentation of the second $K$-group of local rings such that their residue fields have more than four elements. Note that Van der Kallen proved this result when the residue fields have more than five elements. Although we prove our results over local rings, all our proofs also work over semilocal rings where all their residue fields have similar properties as the residue field of local rings.
In this paper, we define the relative higher $rho$ invariant for orientation preserving homotopy equivalence between PL manifolds with boundary in $K$-theory of the relative obstruction algebra, i.e. the relative analytic structure group. We also show that the map induced by the relative higher $rho$ invariant is a group homomorphism from the relative topological structure group to the relative analytic structure group. For this purpose, we generalize Weinberger, Xie and Yus definition of the topological structure group in their article Shmuel Weinberger, Zhizhang Xie, and Guoliang Yu. Additivity of higher rho invariants and nonrigidity of topological manifolds. Communications on Pure and Applied Mathematics, to appear. to make the additive structure of the relative topological structure group transparent.
We introduce the notion of proper Kasparov cycles for Kasparovs G-equivariant KK-theory for a general locally compact, second countable topological group G. We show that for any proper Kasparov cycle, its induced map on K-theory factors through the left-hand side of the Baum-Connes conjecture. This allows us to upgrade the direct splitting method, a recent new approach to the Baum-Connes conjecture which, in contrast to the standard gamma element method (the Dirac dual-Dirac method), avoids the need of constructing proper algebras and the Dirac and the dual-Dirac elements. We introduce the notion of Kasparov cycles with Property (gamma) removing the G-compact assumption on the universal space EG in the previous paper Direct Splitting Method for the Baum-Connes Conjecture. We show that the existence of a cycle with Property (gamma) implies the split-injectivity of the Baum-Connes assembly map for all coefficients. We also obtain results concerning the surjectivity of the assembly map.
Let $X$ be a compact Hausdorff space, let $Gamma$ be a discrete group that acts continuously on $X$ from the right, define $widetilde{X} = {(x,gamma) in X times Gamma : xcdotgamma= x}$, and let $Gamma$ act on $widetilde{X}$ via the formula $(x,gamma)cdotalpha = (xcdotalpha, alpha^{-1}gammaalpha)$. Results of P. Baum and A. Connes, along with facts about the Chern character, imply that $K^i_Gamma(X) otimes mathbb{C} cong K^i(widetilde{X}slashGamma) otimes mathbb{C}$ for $i = 0, -1$. In this note, we present an example where the groups $K^i_Gamma(X)$ and $K^i(widetilde{X}slashGamma)$ are not isomorphic.
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