Alexander-Spanier上循环的指数配对
Index pairing with Alexander-Spanier cocycles
摘要：
本文利用伪微分算子的完备符号代数的循环上同调与K-理论的配对，通过循环同调中的Chern特征标的相对形式，给出了对应于任一奇偶性的Alexander-Spanier上循环的椭圆算子的高阶指数的统一构造.当一个椭圆算子D在一个闭流形M上的最低指数公式（与它的Fredholm指数一致）重现了Atiyah-Singer指数定理时，我们的D的最高指数公式（与体积上圈相关联）产生了Helton-Howe公式在任意维流形上的推广，该公式是关于奇维球面上经典Toeplitz算子的多重算子迹的。事实上，椭圆算子D的高解析指数的总和等于由D确定的K-同调圈的Connes-Chern特征标的表示，其表示形式是将Helton-Howe公式外推到M的维数以下。
英文摘要：
We give a uniform construction of the higher indices of elliptic operators associated to Alexander-Spanier cocycles of either parity in terms of a pairing a la Connes between the K-theory and the cyclic cohomology of the algebra of complete symbols of pseudodifferential operators, implemented by means of a relative form of the Chern character in cyclic homology. While the formula for the lowest index of an elliptic operator D on a closed manifold M (which coincides with its Fredholm index) reproduces the Atiyah-Singer index theorem, our formula for the highest index of D (associated to a volume cocycle) yields an extension to arbitrary manifolds of any dimension of the Helton-Howe formula for the trace of multicommutators of classical Toeplitz operators on odd-dimensional spheres. In fact, the totality of higher analytic indices for an elliptic operator D amount to a representation of the Connes-Chern character of the K-homology cycle determined by D in terms of expressions which extrapolate the Helton-Howe formula below the dimension of M.
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