Topological K-theory for discrete groups and index theory

For any countable discrete group Γ (without further assumptions on it) we first construct an explicit morphism from the Left-Hand side of Baum-Connes assembly map, Ktop⁎(Γ), to periodic cyclic homology algebra, HP⁎(CΓ). This morphism, called Chern-Baum-Connes allows in particular give a proper and formulation for Chern-Connes pairingKtop⁎(Γ)×HP⁎(CΓ)⟶C. Several theorems are needed formulate map. In establish delocalised Riemann-Roch theorem, wrong way functoriality cohomology Γ-proper actions, construction Chern between associated which is isomorphism once tensoring with C, cohomological map H⁎(Γ,FΓ) (where FΓ complex vector space freely generated by set elliptic elements Γ). last identifies, work Burghelea, as direct factor algebra. Moreover, this paper Ktop⁎(Γ) stands topological K-theory groups originally proposed Baum Connes their paper, where equivariant was used its construction. As part our results prove that model indeed isomorphic analytic left-hand Our main final theorem gives geometric index theoretical formula above mentioned pairing Γ) terms pairings invariant forms, cycles given delocalized Todd classes, currents naturally cocycles using Burghelea's computation. complete solution, groups, problem defining computing left hand actions manifolds,