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Zbl 0759.58047
Connes, Alain; Moscovici, Henri
Cyclic cohomology, the Novikov conjecture and hyperbolic groups. (English)
[J] Topology 29, No.3, 345-388 (1990). ISSN 0040-9383

This paper gives a proof of the Novikov conjecture for hyperbolic groups using the techniques of noncommutative differential geometry developed by the first author [Publ. Math., Inst. Hautes Étud. Sci. 62, 257-360 (1985; Zbl 0592.46056)].\par Let $\Gamma$ be a finitely presented group and $\psi$ a continuous map from $M$ to $B\Gamma$. Let $L(M)$ be the Hirzebruch $L$-class of $M$; if $\xi$ is a class in $H\sp*(B\Gamma,\bbfC)$ then the number $\langle L(M)\cdot\psi\sp*(\xi),[M]\rangle$ is a higher signature. The Novikov conjecture says these numbers are oriented homotopy invariants of $(M,\psi)$. For the case in which $\Gamma$ is a discrete subgroup of a Lie group with finitely many components, this conjecture was proved by {\it G. G. Kasparov} using his bivariant $K$-theory [Invent. Math. 91, No. 1, 147-201 (1988; Zbl 0647.46053)]. {\it G. G. Kasparov} and {\it G. Skandalis} extended these techniques to discrete subgroups of $p$-adic and adelic groups [C. R. Acad. Sci., Paris, Sér. I 310, No. 4, 171-174 (1990; Zbl 0705.19010)].\par Let $D$ be an elliptic operator on the compact manifold $M$. The authors show how the Alexander-Spanier cohomology of $M$ naturally pairs with $D$ to yield the localized analytic indices for $D$. They give a cohomological formula for these localized indices using heat equation techniques based on the Getzler calculus of asymptotic pseudodifferential operators. However, the computations are far more intricate than those needed for the Atiyah-Singer index theorem.\par Let $H\sp*(\Gamma,\bbfC)$ be the group cohomology of $\Gamma$, which is isomorphic to $H\sp*(B\Gamma,\bbfC)$. Given $c\in H\sp*(\Gamma,\bbfC)$, we denote by $\xi\sb c$ the corresponding class in $H\sp*(B\Gamma,\bbfC)$. Let $\bbfC\Gamma$ denote the algebraic group ring of $\Gamma$ and $\bbfR$ the algebra of smoothing operators on $L\sp 2(M)$. Given $c\in H\sp*(\Gamma,\bbfC)$, there is a naturally defined cyclic cocycle that gives an additive map $\tau\sb c: K\sb 0(\bbfC\Gamma\otimes R)\to \bbfC$.\par We can view an elliptic operator $D$ on $M$ as a $\Gamma$-invariant operator on the universal cover of $M$; thus, $D$ defines an element $\bar D\to K\sb 0(\bbfC\Gamma\otimes R)$ (this correspondence is made very explicit in the paper). If $c\in H\sp*(\Gamma,\bbfC)$ the authors show that $\langle\tau\sb c,\bar D\rangle=\langle \psi\sp*(\xi\sb c),D\rangle$, the localized analytic index associated with the cohomology class $\psi\sp*(\xi\sb c)\in H\sp*(M,\bbfC)$. In particular, when $D$ is the signature operator on $M$, this yields the higher signature associated to $\psi\sp*(\xi\sb c)$.\par Let $j:\bbfC \Gamma\otimes R\to\bbfC\sp*\sb r(\Gamma)\otimes K(H)$ be the natural inclusion and let $j\sb K: K\sb 0(\bbfC\Gamma\otimes R)\to K(\bbfC\sp*\sb r(\Gamma))$ be the induced map on $K$-theory. Homotopy invariance of the higher signatures follows once one shows that there is a map $\hat \tau\sb c: K(\bbfC\sp*\sb r(\Gamma))\to\bbfC$ such that $\hat\tau\sb c\circ j\sb K=\tau\sb c$. A group cocycle is called extendable if this happens for $\tau\sb c$. It is here that the hyperbolic assumption on the group is used; if $\Gamma$ is hyperbolic then every group cocycle is extendable.\par The authors indicate the possibility of a different approach to the Novikov theorem that avoids the localized analytic indices. There is also an aside on asymptotic cyclic cocycles and a promise of further development of this idea in a future paper.

MSC 2000:: *58J22 Exotic index theories
19D55 K-theory and homology
19K56 Index theory (K-theory)
46L80 K-theory and operator algebras
57R20 Characteristic classes and numbers

Keywords: Novikov conjecture; hyperbolic groups; noncommutative differential geometry

Citations: Zbl 0592.46056; Zbl 0647.46053; Zbl 0705.19010

Cited in: Zbl 1233.20038 Zbl 1188.19003 Zbl 1171.19003 Zbl 1013.20034 Zbl 0982.58018 Zbl 1032.58012 Zbl 0920.58046 Zbl 0906.43009 Zbl 0893.19002 Zbl 0867.19001 Zbl 0868.58076 Zbl 0852.58077 Zbl 0840.57016 Zbl 0819.46058 Zbl 0818.46076 Zbl 0780.58043

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