A geometric approach to K-homology for Lie manifolds
Abstract
We show that the computation of the Fredholm index of a fully elliptic pseudodifferential operator on an integrated Lie manifold can be reduced to the computation of the index of a Dirac operator, perturbed by a smoothing operator, canonically associated, via the so-called clutching map. 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 categorized as a variant of the famous geometric $K$-homology groups, for the specific situation here. We also define a comparison map between this geometric $K$-homology theory and a relative $K$-theory group, directly associated to a fully elliptic pseudodifferential operator.
- Publication:
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arXiv e-prints
- Pub Date:
- April 2019
- DOI:
- 10.48550/arXiv.1904.04069
- arXiv:
- arXiv:1904.04069
- Bibcode:
- 2019arXiv190404069B
- Keywords:
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- Mathematics - K-Theory and Homology;
- Mathematics - Differential Geometry;
- Mathematics - Operator Algebras
- E-Print:
- To appear in Annales de l'ENS