The Atiyah-Patodi-Singer rho invariant and signatures of links

Enrico Toffoli

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arXiv · 2011.00525

The Atiyah-Patodi-Singer rho invariant and signatures of links

Enrico Toffoli

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Abstract

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 $3$-manifold $Y_L$ intrinsically associated to $L$. We study then the rho invariant of the manifolds obtained by Dehn surgery on a $L$ along integer and rational framings. Inspired by related 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.

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Enrico Toffoli. 2020-11-01. The Atiyah-Patodi-Singer rho invariant and signatures of links. https://arxiv.org/abs/2011.00525

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