Bordism, rho-invariants and the Baum-Connes conjecture
Bordism, rho-invariants and the Baum-Connes conjecture
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DOI：
10.4171/jncg/2
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发表时间：
2004-07
影响因子：
0.9
通讯作者：
P. Piazza;T. Schick
中科院分区：
文献类型：
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作者：
P. Piazza;T. Schick
设是一个离散生成群。本文建立了与下列情形相关的rho-不变量的消失结果：（i）具有正数量曲率和基本群的自旋流形的自旋Dirac算子;（ii）具有基本群的同伦等价定向流形对的不交并的签名算子.我们考虑的不变量更准确地说是与一对有限维酉表示相关联的Atiyah-Patodi-Singer（APS）rho不变量1; 2 W！U.d/，Cheeger-Gromov的L2-rho-不变量，Lott的非平凡共轭类的非定域η-不变量。证明了如果极大群C ~*-代数的群存储自由和Baum-Connes映射是双射的，则所有这些rho-不变量都为零.例如，无挠顺从群或SO.n;1/和SU.n;1/的无挠离散子群满足这个条件。对于非定域不变量，我们仅假设Baum-Connes猜想对约化C*-代数成立。除了上面的例子之外，这个条件也可以由Gromov双曲群或SL. 3; C/的余紧离散子群来满足。特别地，与签名算子相关联的三个ρ-不变量对于这样的群是同伦不变量。对于APS和Cheeger-Gromov rho不变量，后者的结果是由Navin Keswani建立的。我们的证明重新建立这一结果，并将其扩展到离域的eta不变的洛特。该证明利用了从边界理论以及APS指数定理的各种推广的基本结果;它还将这些结果嵌入到零度更高的rho-不变量的一般消失现象中（取值于A= λ A; A适合C*-代数A）。我们还获得了关于η-不变量本身的精确信息，这些不变量通常比ρ-不变量更微妙。
Letbe a finitely generated discrete group. In this paper we establish vanishing results for rho-invariants associated to (i) the spin Dirac operator of a spin manifold with positive scalar curvature and fundamental group� ; (ii) the signature operator of the disjoint union of a pair of homotopy equivalent oriented manifolds with fundamental group� . The invariants we consider are more precisely � the Atiyah-Patodi-Singer (� APS) rho-invariant associated to a pair of finite dimensional unitary representations� 1;� 2W� ! U.d/, � theL 2 -rho-invariant of Cheeger-Gromov, � the delocalized eta-invariant of Lott for a non-trivial conjugacy class ofwhich is finite. We prove that all these rho-invariants vanish if the groupistorsion-free and the Baum-Connes map for the maximal group C*-algebra is bijective. This condition is satisfied, for example, by torsion-free amenable groups or by torsion-free discrete subgroups of SO.n;1/ and SU.n;1/. For the delocalized invariant we only assume the validity of the Baum-Connes conjecture for the reduced C*-algebra. In addition to the examples above, this condition is satisfied e.g. by Gromov hyperbolic groups or by cocompact discrete subgroups of SL.3; C/. In particular, the three rho-invariants associated to the signature operator are, for such groups, homotopy invariant. For the APS and the Cheeger-Gromov rho-invariants the latter result had been established by Navin Keswani. Our proof reestablishes this result and also extends it to the delocalized eta-invariant of Lott. The proof exploits in a fundamental way results from bordism theory as well as various generalizations of the APS-index theorem; it also embeds these results in general vanishing phenomena for degree zero higher rho-invariants (taking values in A=ŒA;Afor suitable C*-algebras A). We also obtain precise information about the eta-invariants in question themselves, which are usually much more subtle objects than the rho-invariants.