Counterexamples to the Baum—Connes conjecture
Counterexamples to the Baum—Connes conjecture
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DOI：
10.1007/s00039-002-8249-5
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发表时间：
2002-06
期刊：
影响因子：
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通讯作者：
N. Higson;V. Lafforgue;G. Skandalis
中科院分区：
文献类型：
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作者：
N. Higson;V. Lafforgue;G. Skandalis
Baum-Connes猜想[BC]，[BCH]给出了约化群C <$-代数和叶状C <$-代数的算子K-理论的一个公式.如果G是有限CW-复形的基本群，则G的Baum-Connes猜想可以看作是流形理论中Borel猜想的解析对应，Borel猜想提出了群环Z [G]的L-理论的同调公式。此外，群G的鲍姆-康纳斯猜想实际上暗示了诺维科夫猜想，即具有基本群G的封闭定向流形的高阶签名是定向同伦不变量。由于这个原因，流形理论一直是鲍姆-康纳斯猜想背后的驱动力，而作为回报，算子K理论技术已经证明了一些关于高阶签名同伦不变性的最著名结果。从一开始，对群作用的推广就在鲍姆-康纳斯猜想的发展中发挥了重要作用。最近，进一步的扩展已经被提出到一般的局部紧广群[T1]和粗几何空间[HiR]，[R]。目前的算子代数方法诺维科夫猜想相当严重地依赖于这些（例如见[Hi]）。鲍姆-康纳斯猜想及其推广现在已经在各种情况下得到验证。关于群和群作用的最新工作见[HiK]，[L];关于群胚的工作见[T1];关于粗糙几何空间的工作见[Y]。事实上，现在已经证明的范围是相当显着的，特别是考虑到鲍姆和康纳斯在一开始就提供给他们的信息很少。当然一般的鲍姆-康纳斯猜想是更广泛的，因为它适用于每一个（第二可数）局部紧广群，甚至，在猜想的情况下'与系数'，以每一个行动，这样一个广群对一个C-代数。该猜想有迷人的接触点，不仅与诺维科夫猜想，但与黎曼几何，代表理论的真实的和p进群，和谱理论的离散群。
The Baum–Connes conjecture [BC],[BCH] proposes a formula for the operator K-theory of reduced group C∗-algebras and foliation C∗-algebras. If G is the fundamental group of a finite CW-complex then the Baum–Connes conjecture for G can be viewed as an analytic counterpart of the Borel conjecture in manifold theory, which proposes a homological formula for the L-theory of the group ring Z [G]. Moreover the Baum–Connes conjecture for a group G actually implies Novikov’s conjecture that the higher signatures of a closed, oriented manifold with fundamental group G are oriented homotopy invariants. For this reason manifold theory has been a driving force behind work on the Baum–Connes conjecture, and in return operator K-theory techniques have proved some of the best known results on the homotopy invariance of higher signatures. From the very beginning, generalizations to group actions have played an important role in the development of the Baum–Connes conjecture. More recently, further extensions have been proposed to general locally compact groupoids [T1] and to coarse geometric spaces [HiR],[R]. Current operator algebraic approaches to the Novikov conjecture rely quite heavily on these (see for instance [Hi]).The Baum–Connes conjecture and its generalizations have now been verified in a variety of cases. For recent work on groups and group actions see [HiK],[L]; for work on groupoids see [T1]; and for work on coarse geometric spaces see [Y]. Indeed the scope of what has now been proved is quite remarkable, especially given the scant information which Baum and Connes had available to them at the outset. Of course the general Baum–Connes conjecture is broader still, applying as it does to every (second countable) locally compact groupoid, or even, in the case of the conjecture ‘with coefficients’, to every action of a such a groupoid on a C∗-algebra. The conjecture has fascinating points of contact not only with the Novikov conjecture but with Riemannian geometry, the representation theory of real and p-adic groups, and the spectral theory of discrete groups.