\(K\)-theory, group \(C^*\)-algebras, and higher signatures (conspectus).

*(English)*Zbl 0957.58020
Ferry, Steven C. (ed.) et al., Novikov conjectures, index theorems and rigidity. Vol. I. Based on a conference of the Mathematisches Forschungsinstitut Oberwolfach held in September 1993. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 226, 101-146 (1995).

This fundamental paper is the published version of a preprint by Kasparov, first issued in 1981. It describes the construction of the \(KK\)-groups and the Kasparov product for \(\sigma\)-unital \(C^*\)-algebras equipped with a norm-continuous action of a locally compact group, gives applications to calculating the \(K\)-theory of group algebras of some Lie groups, and proves a strong form of the Novikov conjecture for manifolds with fundamental group \(\pi _1\) that is a discrete subgroup of a suitable Lie group. The Novikov conjecture states that if \(M_1\) and \(M_2\) are closed oriented manifolds, \(h:M_1\to M_2\) is an orientation-preserving homotopy equivalence, and \(t:M _2\to B\pi_1\) is a continuous map into the classifying space of \(\pi_1,\) then
\[
f_*({\mathcal L} (M_2)\cap [M_2])=(f\circ h)_*({\mathcal L} (M_1)\cap [M_1])
\]
in \(H_{*}(B\pi_1,{\mathbf Q}),\) where \({\mathcal L}\) is the Hirzebruch total \({\mathcal L}\)-class.

There is a closely related paper by the same author [Invent. Math. 91, No. 1, 147-201 (1988; Zbl 0647.46053)]. In general, this paper contains more detailed proofs and some technical improvements of the results of the paper under review. The paper under review differs from this other paper in several ways. The paper under review is shorter, less technical, and easier to read. In this paper, the product is constructed using the same approach as in Kasparov’s foundational paper [Izv. Akad. Nauk SSSR, Ser. Mat. 44, 571-636 (Russian) (1980; Zbl 0448.46051)], not the connection approach due to A. Connes and G. Skandalis [Publ. Res. Inst. Math. Sci. 20, 1139-1183 (1984; Zbl 0575.58030)] that is used in the Inventiones paper.

Section 5 of the conspectus gives a careful and detailed description of the Kasparov \(\gamma\)-element, which is given by a pairing between the Dirac element and the “dual Dirac” element. These elements are equivariant versions of the Bott and dual Bott elements that appear in classical \(K\)-theory and \(K\)-homology. The gamma element is an idempotent in the Kasparov representation ring of the group that is acting, and the Novikov conjecture can be shown to hold in the cases where \(\gamma =1 \). This section also contains results about the behaviour of the \(KK_\Gamma\) groups under change of group, and the construction of the induction homomorphisms \(j^G:KK^{G\times\Gamma}(A,B)\to KK^\Gamma) (C^*(G,A)\), \(C^*(G,B))\). In Section 6, there are some results not given in the Izvestiya paper, about the exact sequence \(0\to B\to C^*(G)\to C^*(G/ \Gamma) \to 0\), where \(\Gamma \triangleleft G\) is a normal subgroup of the nilpotent Lie group \(G\) and \(B\) is a noncommutative generalization of \(C(X) \otimes {\mathcal K} \), where \(X\) is a certain base space. The results are that the sequence does not split, and it is identified with a particular element of the group of extensions \(KK^1(C^*(G/ \Gamma),B)\). Section 7 studies the \(\gamma\)-element for algebras coming from discrete groups. In Section 8, the Kasparov \(\gamma\)-element is used to construct a PoincarĂ© duality for complete Riemannian manifolds. In Section 9 there is an explanation of how to reduce the question of homotopy invariance of the higher signatures to a problem in \(K\)-theory of the classifying space of the fundamental group \(\pi_1\). This \(K\)-theoretical strong form of the Novikov conjecture is proven for the case where \(\pi_1\) is a discrete subgroup of a connected Lie group.

I enthusiastically recommend this clearly written paper to anyone interested in Kasparov’s approach to the Novikov conjecture, or in \(KK\)-theory.

For the entire collection see [Zbl 0829.00027].

There is a closely related paper by the same author [Invent. Math. 91, No. 1, 147-201 (1988; Zbl 0647.46053)]. In general, this paper contains more detailed proofs and some technical improvements of the results of the paper under review. The paper under review differs from this other paper in several ways. The paper under review is shorter, less technical, and easier to read. In this paper, the product is constructed using the same approach as in Kasparov’s foundational paper [Izv. Akad. Nauk SSSR, Ser. Mat. 44, 571-636 (Russian) (1980; Zbl 0448.46051)], not the connection approach due to A. Connes and G. Skandalis [Publ. Res. Inst. Math. Sci. 20, 1139-1183 (1984; Zbl 0575.58030)] that is used in the Inventiones paper.

Section 5 of the conspectus gives a careful and detailed description of the Kasparov \(\gamma\)-element, which is given by a pairing between the Dirac element and the “dual Dirac” element. These elements are equivariant versions of the Bott and dual Bott elements that appear in classical \(K\)-theory and \(K\)-homology. The gamma element is an idempotent in the Kasparov representation ring of the group that is acting, and the Novikov conjecture can be shown to hold in the cases where \(\gamma =1 \). This section also contains results about the behaviour of the \(KK_\Gamma\) groups under change of group, and the construction of the induction homomorphisms \(j^G:KK^{G\times\Gamma}(A,B)\to KK^\Gamma) (C^*(G,A)\), \(C^*(G,B))\). In Section 6, there are some results not given in the Izvestiya paper, about the exact sequence \(0\to B\to C^*(G)\to C^*(G/ \Gamma) \to 0\), where \(\Gamma \triangleleft G\) is a normal subgroup of the nilpotent Lie group \(G\) and \(B\) is a noncommutative generalization of \(C(X) \otimes {\mathcal K} \), where \(X\) is a certain base space. The results are that the sequence does not split, and it is identified with a particular element of the group of extensions \(KK^1(C^*(G/ \Gamma),B)\). Section 7 studies the \(\gamma\)-element for algebras coming from discrete groups. In Section 8, the Kasparov \(\gamma\)-element is used to construct a PoincarĂ© duality for complete Riemannian manifolds. In Section 9 there is an explanation of how to reduce the question of homotopy invariance of the higher signatures to a problem in \(K\)-theory of the classifying space of the fundamental group \(\pi_1\). This \(K\)-theoretical strong form of the Novikov conjecture is proven for the case where \(\pi_1\) is a discrete subgroup of a connected Lie group.

I enthusiastically recommend this clearly written paper to anyone interested in Kasparov’s approach to the Novikov conjecture, or in \(KK\)-theory.

For the entire collection see [Zbl 0829.00027].

Reviewer: Daniel Kucerovsky (MR 97j:58153)