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- W2947488508 abstract "Let $Gamma$ be a finitely generated discrete group and let $widetilde{M}$ be a Galois $Gamma$-covering of a smooth compact manifold $M$. Let $u:Xto BGamma$ be the associated classifying map. Finally, let $mathrm{S}_*^Gamma (widetilde{M})$ be the analytic structure group, a K-theory group appearing in the Higson-Roe exact sequence $cdotsto mathrm{S}_*^Gamma(widetilde{M})to K_*(M)to K_*(C^*Gamma)tocdots$. Under suitable assumptions on $Gamma$ we construct two pairings, first between $mathrm{S}^Gamma_*(widetilde{M})$ and the delocalized part of the cyclic cohomology of $mathbb{C}Gamma$, and secondly between $mathrm{S}^Gamma_*(widetilde M)$ and the relative cohomology $H^*(Mto BGamma)$. Both are compatible with known pairings associated with the other terms in the Higson-Roe sequence. In particular, we define higher rho numbers associated to the rho class $rho(widetilde{D})in mathrm{S}_*^Gamma (widetilde{M})$ of an invertible $Gamma$-equivariant Dirac type operator on $widetilde{M}$. Regarding the first pairing we establish in fact a more general result, valid without additional assumptions on $Gamma$: indeed, we prove that it is possible to map the Higson-Roe analytic surgery sequence to the long exact sequence in noncommutative de Rham homology $ cdotsxrightarrow{j_*} H_{*-1} (mathcal{A}Gamma)xrightarrow{iota} H^{del}_{*-1} (mathcal{A}Gamma)xrightarrow{delta}H^{e}_{*} (mathcal{A}Gamma)xrightarrow{j_*}cdots$ with $mathcal{A}Gamma$ a dense homomorphically closed subalgebra of $C^*_r Gamma$ and $ H_{*}^{del} (mathcal{A}Gamma)$ and $ H_{*}^{e} (mathcal{A}Gamma)$ denoting versions of the delocalized homology and the homology localized at the identity element, respectively." @default.
- W2947488508 created "2019-06-07" @default.
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- W2947488508 date "2019-05-28" @default.
- W2947488508 modified "2023-09-27" @default.
- W2947488508 title "Higher rho numbers and the mapping of analytic surgery to homology" @default.
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