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The Atiyah–Patodi–Singer rho invariant and signatures of links
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 65 / Issue 2 / May 2022
- Published online by Cambridge University Press:
- 22 April 2022, pp. 404-440
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- Article
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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.
