Connes-Chern character for manifolds with boundary and eta cochains
Connes-Chern character for manifolds with boundary and eta cochains
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具有边界和 eta 上链的流形的 Connes-Chern 特征
DOI：
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发表时间：
2009
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通讯作者：
M. Pflaum
中科院分区：
文献类型：
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作者：
M. Lesch;H. Moscovici;M. Pflaum
我们用相对循环上同调中的收缩上圈来表示与具有边界的流形上的b-度量有关的Dirac算子的Connes-Chern特征，它的表示依赖于标度/截断参数。吹起度规1恢复了代表相应的De Rham相对同调类的特征电流对，而吹倒产生了一个相对共循环，其表达涉及更高的ETA协链及其b-类似物。与相对K-理论类对应的配对公式捕捉了关于边界的信息，并允许导出几何推论。作为副产品，我们证明了由Getzler和Wu引入的广义Atiyah-Patodi-Singer配对必然限于几乎平坦的丛。
We express the Connes-Chern character of the Dirac operator associated to a b-metric on a manifold with boundary in terms of a retracted cocycle in relative cyclic cohomology, whose expression depends on a scaling/cut-off pa- rameter. Blowing-up the metric one recovers the pair of characteristic currents that represent the corresponding de Rham relative homology class, while the blow-down yields a relative cocycle whose expression involves higher eta cochains and their b-analogues. The corresponding pairing formulae with relative K-theory classes capture information about the boundary and allow to derive geometric consequences. As a by-product, we show that the generalized Atiyah-Patodi-Singer pairing introduced by Getzler and Wu is necessarily restricted to almost flat bundles.