Poisson maps between character varieties: gluing and capping

TL;DR

This paper investigates Poisson maps between G-character varieties induced by surface mappings, providing an algorithm for computing Poisson bi-vectors, exemplified on a 5-holed sphere with Euler characteristic -3.

Contribution

It establishes that these induced maps are generally Poisson and introduces an effective algorithm for computing Poisson bi-vectors for G=SL(2,C).

Findings

Mappings are generally Poisson.

Algorithm successfully computes Poisson bi-vectors.

Explicit example for a 5-holed sphere provided.

Abstract

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=SL(2,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.