Search

Relations between the Atiyah–Patodi–Singer rho invariant and signatures of links have been known for a long time, but they were only partially investigated. In order to explore them further, we develop a versatile cut-and-paste formula for the rho invariant, which allows us to manipulate manifolds in a convenient way. With the help of this tool, we give a description of the multivariable signature of a link $L$ as the rho invariant of some closed three-manifold $Y_L$ intrinsically associated with $L$. We study then the rho invariant of the manifolds obtained by the Dehn surgery on $L$ along integer and rational framings. Inspired by the results of Casson and Gordon and Cimasoni and Florens, we give formulas expressing this value as a sum of the multivariable signature of $L$ and some easy-to-compute extra terms. [ABSTRACT FROM AUTHOR]

Let M be a closed spin manifold which supports a positive scalar curvature metric. The set of concordance classes of positive scalar curvature metrics on M forms an abelian group P (M) after fixing a positive scalar curvature metric. The group P (M) measures the size of the space of positive scalar curvature metrics on M. Weinberger and Yu gave a lower bound of the rank of P (M) in terms of the number of torsion elements of π 1 (M). In this paper, we give a sharper lower bound of the rank of P (M) by studying the image of the relative higher index map from P (M) to the real K-theory of the group C ⁎ -algebra C r ⁎ (π 1 (M)). We show that it rationally contains the image of the Baum–Connes assembly map up to a certain homological degree depending on the dimension of M. At the same time we obtain lower bounds for the positive scalar curvature bordism group by applying the higher rho-invariant. [ABSTRACT FROM AUTHOR]