论文标题
$ l^1 $ - 高索引，$ l^1 $ - 高rho不变和环状同胞
$l^1$-higher index, $l^1$-higher rho invariant and cyclic cohomology
论文作者
论文摘要
在本文中，我们研究了$ l^1 $ - 高索引理论及其与循环的共同体配对，以构成封闭的歧管和带边界的紧凑型歧管。我们首先给出了足够的几何条件，以消除封闭的歧管上的dirac型运算符的$ l^1 $ - 更高的索引。这导致我们定义了$ l^1 $ rho不变性的。我们证明了这些$ l^1 $ - 高的Rho不变性的产品公式。我们产品公式的主要新颖性是它在总Banach代数环境中起作用，尤其是$ l^1 $ setting。 在带边界的紧凑型自旋歧管上，我们还提供了足够的几何条件，使狄拉克操作员具有明确的$ l^1 $ - 更高的索引。更准确地说，我们表明，在紧凑的旋转歧管上，边界上配备了一个带有Riemannian指标的边界，如果边界上的标量曲率足够大，则具有$ l^1 $ - 高的d_m $ $ d_m $的索引，然后在$ k $ $ l^$ l^$ l^$ l^$ l^$ l^$ l^$ l^1中。作为立即推论，我们会看到，如果BOST的猜想是$ M $的基本组，则$ c^\ ast $ -Algebraic $ d_m $ $ $ $ $ $ $ d_m $的高指数在于Baum-Connes组装图的图像。 通过将上述$ k $ - 理论$ l^1 $ index结果与循环共体配对，我们证明了$ l^1 $ - 较高的Atiyah-patodi-singer索引定理具有边界的流形。其证明的关键要素是上述$ l^1 $高的Rho不变性的产品公式。
In this paper, we study $l^1$-higher index theory and its pairing with cyclic cohomology for both closed manifolds and compact manifolds with boundary. We first give a sufficient geometric condition for the vanishing of the $l^1$-higher indices of Dirac-type operators on closed manifolds. This leads us to define an $l^1$-version of higher rho invariants. We prove a product formula for these $l^1$-higher rho invariants. A main novelty of our product formula is that it works in the general Banach algebra setting, in particular, the $l^1$-setting. On compact spin manifolds with boundary, we also give a sufficient geometric condition for Dirac operators to have well-defined $l^1$-higher indices. More precisely, we show that, on a compact spin manifold $M$ with boundary equipped with a Riemannian metric which has product structure near the boundary, if the scalar curvature on the boundary is sufficiently large, then the $l^1$-higher index of its Dirac operator $D_M$ is well-defined and lies in the $K$-theory of the $l^1$-algebra of the fundamental group. As an immediate corollary, we see that if the Bost conjecture holds for the fundamental group of $M$, then the $C^\ast$-algebraic higher index of $D_M$ lies in the image of the Baum-Connes assembly map. By pairing the above $K$-theoretic $l^1$-index results with cyclic cocycles, we prove an $l^1$-version of the higher Atiyah-Patodi-Singer index theorem for manifolds with boundary. A key ingredient of its proof is the product formula for $l^1$-higher rho invariants mentioned above.
