2,241 research outputs found
K-duality for stratified pseudomanifolds
This paper is devoted to the study of Poincar\'e duality in K-theory for
general stratified pseudomanifolds. We review the axiomatic definition of a
smooth stratification \fS of a topological space and we define a groupoid
T^{\fS}X, called the \fS-tangent space. This groupoid is made of different
pieces encoding the tangent spaces of the strata, and these pieces are glued
into the smooth noncommutative groupoid T^{\fS}X using the familiar procedure
introduced by A. Connes for the tangent groupoid of a manifold. The main result
is that C^{*}(T^{\fS}X) is Poincar\'e dual to , in other words, the
\fS-tangent space plays the role in -theory of a tangent space for
A geometric approach to K-homology for Lie manifolds
We prove that the computation of the Fredholm index for fully elliptic
pseudodifferential operators on Lie manifolds can be reduced to the computation
of the index of Dirac operators perturbed by smoothing operators. To this end
we adapt to our framework ideas coming from Baum-Douglas geometric K-homology
and in particular we introduce a notion of geometric cycles that can be
classified into a variant of the famous geometric K-homology groups, for the
specific situation here. We also define comparison maps between this geometric
K-homology theory and relative K-theory
Fourier integrals operators on lie groupoids
As announced in [12], we develop a calculus of Fourier integral G-operators
on any Lie groupoid G. For that purpose, we study convolability and
invertibility of Lagrangian conic submanifolds of the symplectic groupoid T *
G. We also identify those Lagrangian which correspond to equivariant families
parametrized by the unit space G (0) of homogeneous canonical relations in (T *
Gx \ 0) x (T * G x \ 0). This allows us to select a subclass of Lagrangian
distributions on any Lie groupoid G that deserve the name of Fourier integral
G-operators (G-FIO). By construction, the class of G-FIO contains the class of
equivariant families of ordinary Fourier integral operators on the manifolds
Gx, x G (0). We then develop for G-FIO the first stages of the calculus
in the spirit of Hormander's work. Finally, we work out an example proving the
efficiency of the present approach for studying Fourier integral operators on
singular manifolds
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