Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Topological K-theory for discrete groups and Index theory   We give a complete solution, for discrete countable groups, to the problem of
defining and computing a geometric pairing between the left hand side of the
Baum-Connes assembly map, given in terms of geometric cycles associated to
proper actions on manifolds, and cyclic periodic cohomology of the group
algebra. Indeed, for any such group $\Gamma$ (without any further assumptions
on it) we construct an explicit morphism from the Left-Hand side of the
Baum-Connes assembly map to the periodic cyclic homology of the group algebra.
This morphism, called here the Chern-Baum-Connes assembly map, allows to give a
proper and explicit formulation for a Chern-Connes pairing with the periodic
cyclic cohomology of the group algebra.
  Several theorems are needed to formulate the Chern-Baum-Connes assembly map.
In particular we establish a delocalised Riemann-Roch theorem, the wrong way
functoriality for periodic delocalised cohomology for $\Gamma$-proper actions,
the construction of a Chern morphism between the Left-Hand side of Baum-Connes
and a delocalised cohomology group associated to $\Gamma$ which is an
isomorphism once tensoring with $\mathbb{C}$, and the construction of an
explicit cohomological assembly map between the delocalised cohomology group
associated to $\Gamma$ and the homology group $H_*(\Gamma,F\Gamma)$.
  We then give an index theoretical formula for the above mentioned pairing
(for any $\Gamma$) in terms of pairings of invariant forms, associated to
geometric cycles and given in terms of delocalized Chern and Todd classes, and
currents naturally associated to group cocycles using Burghelea's computation.
As part of our results we prove that left-Hand side group used in this paper is
isomorphic to the usual analytic model for the left-hand side of the assembly
map.

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The categories of this paper are: hep-th
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The categories of this paper are: cs
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The categories of this paper are: chem-ph
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The categories of this paper are: econ
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The categories of this paper are: atom-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: plasm-ph
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The categories of this paper are: q-alg
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The categories of this paper are: patt-sol
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The categories of this paper are: nlin
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The categories of this paper are: physics
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The categories of this paper are: alg-geom
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The categories of this paper are: mtrl-th
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The categories of this paper are: q-bio
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The categories of this paper are: quant-ph
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The categories of this paper are: eess