The Novikov conjecture. (English. Russian original) Zbl 1439.19009
Russ. Math. Surv. 74, No. 3, 525-541 (2019); translation from Usp. Mat. Nauk 74, No. 3, 167-184 (2019).
In the current pure mathematical universe, Novikov Conjecture is one of the central problems. Novikov Conjecture has its root to the Hirzebruch signature theorem, which states that the signature of an oriented closed \(4k\)-dimensional manifold in terms of characteristic classes can be expressed by a linear combination of Pontryagin numbers called the \(L\)-genus. Novikov Conjecture states that higher signatures are homotopy invariants for all orientation-preserving homotopy equivalences between closed oriented \(n\)-dimensional manifolds.
This survey concentrates on “the analytic approach based on ideas from noncommutative geometry”. “The Novikov Conjecture follows from the rational strong Novikov Conjecture, which states that the rational Kasparov-Baum-Connes map is injective, where the Kasparov-Baum-Connes map is the assembly map from the \(K\)-homology of the classifying space for the fundamental group to the \(K\)-theory of the group \(C^*\)-algebra associated to the fundamental group.” Along this path, a long list of groups that have been proved for the Novikov Conjecture is summarized. The author puts them into several categories.
The first class of groups is associated with various non-positive curvature conditions. The key words are non-positively curved groups, hyperbolic groups and bolic groups. Connes’ cyclic cohomology theory, Kasparov’s \(KK\)-theory, Gromov’s hyperbolic group theory were developed in order to prove some of the results.
The second class of groups is associated to asymptotic dimensions, Yu’s Property A, Hilbert spaces and Banach spaces. In the research of this class, a lot of beautiful theories were developed, such as Higson-Kasparov’s \(E\)-theory, Yu’s quantitative operator \(K\)-theory, Yu’s localization algebra. This is an expeditiously expanding area recently. Various inspiring concepts and techniques were developed in the last two decades. Kasparov-Yu’s Property \(H\) of Banach spaces is discussed which is used to generalize the proof from the Hilbert space case to certain Banach space cases. Several open questions are included in this section.
In the recent work by Gong-Wu-Yu, the authors proved that the Novikov Conjecture holds for groups acting properly and isometrically on infinite dimensional non-positively curved spaces. In particular, this implies the Novikov Conjecture for geometrically discrete subgroups of the group of volume preserving diffeomorphisms.
In the final part, the author introduces geometric complexity, a generalization of asymptotic dimensions, and discussed the work of Guentner-Tessera-Yu on the stable Borel Conjecture, a fundamental problem on rigidity of manifolds.
This survey concentrates on “the analytic approach based on ideas from noncommutative geometry”. “The Novikov Conjecture follows from the rational strong Novikov Conjecture, which states that the rational Kasparov-Baum-Connes map is injective, where the Kasparov-Baum-Connes map is the assembly map from the \(K\)-homology of the classifying space for the fundamental group to the \(K\)-theory of the group \(C^*\)-algebra associated to the fundamental group.” Along this path, a long list of groups that have been proved for the Novikov Conjecture is summarized. The author puts them into several categories.
The first class of groups is associated with various non-positive curvature conditions. The key words are non-positively curved groups, hyperbolic groups and bolic groups. Connes’ cyclic cohomology theory, Kasparov’s \(KK\)-theory, Gromov’s hyperbolic group theory were developed in order to prove some of the results.
The second class of groups is associated to asymptotic dimensions, Yu’s Property A, Hilbert spaces and Banach spaces. In the research of this class, a lot of beautiful theories were developed, such as Higson-Kasparov’s \(E\)-theory, Yu’s quantitative operator \(K\)-theory, Yu’s localization algebra. This is an expeditiously expanding area recently. Various inspiring concepts and techniques were developed in the last two decades. Kasparov-Yu’s Property \(H\) of Banach spaces is discussed which is used to generalize the proof from the Hilbert space case to certain Banach space cases. Several open questions are included in this section.
In the recent work by Gong-Wu-Yu, the authors proved that the Novikov Conjecture holds for groups acting properly and isometrically on infinite dimensional non-positively curved spaces. In particular, this implies the Novikov Conjecture for geometrically discrete subgroups of the group of volume preserving diffeomorphisms.
In the final part, the author introduces geometric complexity, a generalization of asymptotic dimensions, and discussed the work of Guentner-Tessera-Yu on the stable Borel Conjecture, a fundamental problem on rigidity of manifolds.
Reviewer: Lin Shan (San Juan)
MSC:
| 19K56 | Index theory |
| 57R20 | Characteristic classes and numbers in differential topology |
| 19K35 | Kasparov theory (\(KK\)-theory) |
| 20F65 | Geometric group theory |
| 57M07 | Topological methods in group theory |
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