The higher index of the signature operator is a far-reaching generalization of the 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 the higher index and the higher rho invariant of the signature operator on a fibered manifold. Our result implies the classical product formula for the numerical signature of a fibered manifold obtained by Chern, Hirzebruch, and Serre (1957). We also give a new proof of the product formula for the higher rho invariant of the signature operator on a product manifold, which is parallel to the product formula for the higher rho invariant of Dirac operator on a product manifold obtained by Xie and Yu (2014) and Zeidler (2016).

中文翻译：

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关于纤维流形的局部特征和更高的rho不变量

签名算子的较高指数是封闭定向流形签名的深远概括。当两个封闭定向流形同伦等价时，可以定义相对签名算子的二级不变量，称为更高的 rho 不变量。较高的 rho 不变量检测流形的拓扑非刚性。在本文中，我们证明了纤维流形上签名算子的更高指数和更高rho不变量的乘积公式。我们的结果暗示了由 Chern、Hirzebruch 和 Serre (1957) 获得的纤维流形的数字签名的经典乘积公式。我们还给出了乘积流形上签名算子的更高rho不变量的乘积公式的新证明，