##
**Cyclic cohomology, the Novikov conjecture and hyperbolic groups.**
*(English)*
Zbl 0759.58047

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)].

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 \(H^*(B\Gamma,\mathbb{C})\) then the number \(\langle L(M)\cdot\psi^*(\xi),[M]\rangle\) 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 G. G. Kasparov using his bivariant \(K\)-theory [Invent. Math. 91, No. 1, 147-201 (1988; Zbl 0647.46053)]. G. G. Kasparov and 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)].

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.

Let \(H^*(\Gamma,\mathbb{C})\) be the group cohomology of \(\Gamma\), which is isomorphic to \(H^*(B\Gamma,\mathbb{C})\). Given \(c\in H^*(\Gamma,\mathbb{C})\), we denote by \(\xi_ c\) the corresponding class in \(H^*(B\Gamma,\mathbb{C})\). Let \(\mathbb{C}\Gamma\) denote the algebraic group ring of \(\Gamma\) and \(\mathbb{R}\) the algebra of smoothing operators on \(L^ 2(M)\). Given \(c\in H^*(\Gamma,\mathbb{C})\), there is a naturally defined cyclic cocycle that gives an additive map \(\tau_ c: K_ 0(\mathbb{C}\Gamma\otimes R)\to \mathbb{C}\).

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 \(\bar D\to K_ 0(\mathbb{C}\Gamma\otimes R)\) (this correspondence is made very explicit in the paper). If \(c\in H^*(\Gamma,\mathbb{C})\) the authors show that \(\langle\tau_ c,\bar D\rangle=\langle \psi^*(\xi_ c),D\rangle\), the localized analytic index associated with the cohomology class \(\psi^*(\xi_ c)\in H^*(M,\mathbb{C})\). In particular, when \(D\) is the signature operator on \(M\), this yields the higher signature associated to \(\psi^*(\xi_ c)\).

Let \(j:\mathbb{C} \Gamma\otimes R\to\mathbb{C}^*_ r(\Gamma)\otimes K(H)\) be the natural inclusion and let \(j_ K: K_ 0(\mathbb{C}\Gamma\otimes R)\to K(\mathbb{C}^*_ r(\Gamma))\) be the induced map on \(K\)-theory. Homotopy invariance of the higher signatures follows once one shows that there is a map \(\hat \tau_ c: K(\mathbb{C}^*_ r(\Gamma))\to\mathbb{C}\) such that \(\hat\tau_ c\circ j_ K=\tau_ c\). A group cocycle is called extendable if this happens for \(\tau_ c\). It is here that the hyperbolic assumption on the group is used; if \(\Gamma\) is hyperbolic then every group cocycle is extendable.

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.

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 \(H^*(B\Gamma,\mathbb{C})\) then the number \(\langle L(M)\cdot\psi^*(\xi),[M]\rangle\) 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 G. G. Kasparov using his bivariant \(K\)-theory [Invent. Math. 91, No. 1, 147-201 (1988; Zbl 0647.46053)]. G. G. Kasparov and 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)].

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.

Let \(H^*(\Gamma,\mathbb{C})\) be the group cohomology of \(\Gamma\), which is isomorphic to \(H^*(B\Gamma,\mathbb{C})\). Given \(c\in H^*(\Gamma,\mathbb{C})\), we denote by \(\xi_ c\) the corresponding class in \(H^*(B\Gamma,\mathbb{C})\). Let \(\mathbb{C}\Gamma\) denote the algebraic group ring of \(\Gamma\) and \(\mathbb{R}\) the algebra of smoothing operators on \(L^ 2(M)\). Given \(c\in H^*(\Gamma,\mathbb{C})\), there is a naturally defined cyclic cocycle that gives an additive map \(\tau_ c: K_ 0(\mathbb{C}\Gamma\otimes R)\to \mathbb{C}\).

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 \(\bar D\to K_ 0(\mathbb{C}\Gamma\otimes R)\) (this correspondence is made very explicit in the paper). If \(c\in H^*(\Gamma,\mathbb{C})\) the authors show that \(\langle\tau_ c,\bar D\rangle=\langle \psi^*(\xi_ c),D\rangle\), the localized analytic index associated with the cohomology class \(\psi^*(\xi_ c)\in H^*(M,\mathbb{C})\). In particular, when \(D\) is the signature operator on \(M\), this yields the higher signature associated to \(\psi^*(\xi_ c)\).

Let \(j:\mathbb{C} \Gamma\otimes R\to\mathbb{C}^*_ r(\Gamma)\otimes K(H)\) be the natural inclusion and let \(j_ K: K_ 0(\mathbb{C}\Gamma\otimes R)\to K(\mathbb{C}^*_ r(\Gamma))\) be the induced map on \(K\)-theory. Homotopy invariance of the higher signatures follows once one shows that there is a map \(\hat \tau_ c: K(\mathbb{C}^*_ r(\Gamma))\to\mathbb{C}\) such that \(\hat\tau_ c\circ j_ K=\tau_ c\). A group cocycle is called extendable if this happens for \(\tau_ c\). It is here that the hyperbolic assumption on the group is used; if \(\Gamma\) is hyperbolic then every group cocycle is extendable.

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.