Journal of Symplectic Geometry

Volume 20 (2022)

Number 6

Poisson maps between character varieties: gluing and capping

Pages: 1255 – 1312



Indranil Biswas (School of Mathematics, Tata Institute of Fundamental Research, Mumbai, India)

Jacques Hurtubise (Department of Mathematics, McGill University, Montreal, Quebec, Canada)

Lisa C. Jeffrey (Department of Mathematics, University of Toronto, Ontario, Canada)

Sean Lawton (Department of Mathematical Sciences, George Mason University, Fairfax, Virginia, U.S.A.)


Let $G$ be a compact Lie group or a complex reductive affine algebraic group. We explore induced mappings between $G$-character varieties of surface groups by mappings between corresponding surfaces. It is shown that these mappings are generally Poisson. We also given an effective algorithm to compute the Poisson bi-vectors when $G = \mathrm{SL} (2, \mathbb{C})$. We demonstrate this algorithm by explicitly calculating the Poisson bi-vector for the $5$-holed sphere, the first example for an Euler characteristic $-3$ surface.

Received 12 April 2021

Accepted 19 April 2022

Published 26 April 2023