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The Novikov conjecture for groups with finite asymptotic dimension. (English) Zbl 0911.19001

The main result is the confirmation of the coarse Baum-Connes conjecture (BBC) for metric spaces with finite asymptotic dimension (FAD).
Coarse geometry has become an important notion in index theory. In particular: the coarse BBC postulates that a certain index map defines an isomorphism between the \(K\)-homology of a space \(X\) to the \(K\)-homology of the \(C^*\)-algebra associated to \(X\). If \(\Gamma\) is a finitely generated group such that its classifying space \(B\Gamma\) has the homotopy type of a finite CW-complex, the coarse BCC for \(\Gamma\) (with word-length metric) implies the strong Novikov conjecture (NC) for \(\Gamma\), that is another index map from the \(K\)-homology of the classifying space \(B\Gamma\) to the \(K\)-homology of the reduced \(C^*\)-algebra of \(\Gamma\) is injective: coarse BCC\(\Rightarrow\)strong NC. This is the \(C^*\)-algebra version of the decent principle [J. Roe, “Coarse cohomology and index theory on complete Riemannian manifolds”, Mem. Am. Math. Soc. 497 (1993; Zbl 0780.58043)], previously known to topologists.
Hence, the main theorem implies the NC for finitely generated groups \(\Gamma\) such that \(B\Gamma\) has the homotopy type of a finite CW-complex and \(\Gamma\) with the word-length metric has FAD.
Asymptotic dimension is a coarse analogue of covering dimension. It is known that Gromov hyperbolic groups have FAD and the author proves that the property FAD is inherited by subgroups. The assumption FAD might be weaker than the assumption that \(B\Gamma\) is of finite type as no example of a group with infinite asymptotic dimension and finite classifying space is known. On the other hand, FAD is a necessary assumption for the coarse BCC; the author constructs an easy example with infinite asymptotic dimension for which the BCC fails.
By previous results of the author, the main result implies also the Gromov-Lawson-Rosenberg conjecture for groups \(\Gamma\) with FAD and finite \(B\Gamma\), as well as Gromov’s zero-in-the-spectrum conjecture for uniformly contractible Riemannian manifolds with FAD.
The main new tool is what the author calls controlled operator \(K\)-theory. In the central part of the paper he constructs certain controlled versions of the \(K\)-theory obstruction groups to the coarse BCC and establishes a Mayer-Vietoris type exact sequence for these. The final argument is an estimate of these obstruction groups for the nerve spaces \(N_{C_n}\) of larger and larger covers \(C_n\) (anti-Čech systems) of the space \(X\) with FAD, which vanish in the limit as \(n\to\infty\).

MSC:

19K56 Index theory
46L80 \(K\)-theory and operator algebras (including cyclic theory)
58J22 Exotic index theories on manifolds
46L85 Noncommutative topology

Citations:

Zbl 0780.58043
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