关于纤维流形的局部化特征标和高ρ不变量
On localized signature and higher rho invariant of fibered manifolds
摘要：
签名算子的高指数是闭定向流形签名算子的一个广泛推广。当两个闭定向流形同伦等价时，可以定义相对签名算子的一个次不变量，称为高ρ不变量。高ρ不变量检测流形的拓扑非刚性。本文证明了纤维流形上签名算子的高指标和高ρ不变量的乘积公式。我们的结果包含了Chern，Hirzebruch，和Serre在“On the index of a fibered manifold”中得到的关于纤维流形的数字签名的经典乘积公式。给出了乘积流形上签名算子的高rho不变量的乘积公式的新证明，它与谢和余在“正数量曲率、高rho不变量和局部化代数”以及Zeidler在“正数量曲率和次指数不变量的乘积公式”中给出的乘积流形上Dirac算子的高rho不变量的乘积公式是平行的.
英文摘要：
Higher index of signature operator is a far reaching generalization of signature of a closed oriented manifold. When two closed oriented manifolds are homotopy equivalent, one can define a secondary invariant of the relative signature operator called higher rho invariant. The higher rho invariant detects the topological nonrigidity of a manifold. In this paper, we prove product formulas for higher index and higher rho invariant of signature operator on fibered manifolds. Our result implies the classical product formula for numerical signature of fiber manifolds obtained by Chern, Hirzebruch, and Serre in "On the index of a fibered manifold". We also give a new proof of the product formula for higher rho invariant of signature operator on product manifolds, which is parallel to the product formula for higher rho invariant of Dirac operator on product manifolds obtained by Xie and Yu in "Positive scalar curvature, higher rho invariants and localization algebras" and Zeidler in "Positive scalar curvature and product formulas for secondary index invariants".
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