The index of G-transversally elliptic families I
Alexandre Baldare

TL;DR
This paper develops an index theory for families of G-transversally elliptic operators, extending Atiyah-Singer results, and computes their Kasparov intersection product using functorial properties and advanced operator algebra techniques.
Contribution
It introduces a new index map for G-transversally elliptic families and establishes its fundamental properties, extending classical index theory to a broader setting.
Findings
Defined the index map for G-transversally elliptic families.
Proved axiomatic properties extending Atiyah-Singer index theorem.
Computed Kasparov intersection product for the index class.
Abstract
We define and study the index map for families of -transversally elliptic operators and introduce the multiplicity for a given irreducible representation as a virtual bundle over the base of the fibration. We then prove the usual axiomatic properties for the index map extending the Atiyah-Singer results [1]. Finally, we compute the Kasparov intersection product of our index class against the K-homology class of an elliptic operator on the base. Our approach is based on the functorial properties of the intersection product, and relies on some constructions due to Connes-Skandalis and to Hilsum-Skandalis.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Operator Algebra Research · Homotopy and Cohomology in Algebraic Topology
