Modules, fields of definition, and the Culler-Shalen norm
Charles Katerba
TL;DR
This paper explores the algebraic structures of character varieties in 3-manifolds, deriving a formula linking module ranks, fields of definition, and the Culler-Shalen norm to aid in understanding essential surfaces.
Contribution
It introduces a new formula for the rank of modules associated with character varieties, connecting algebraic and geometric properties of 3-manifolds.
Findings
Derived a formula for module rank involving the field of definition and Culler-Shalen norm.
Established conditions under which these modules are finitely generated.
Linked algebraic module properties to geometric essential surface construction.
Abstract
Culler-Shalen theory uses the algebraic geometry of the SL(2,C)-character variety of a 3-manifold to construct essential surfaces in the manifold. There are module structures associated to the coordinate rings of the irreducible components of character varieties that are intimately related to essential surface construction. When these modules are finitely generated, we derive a formula for their rank that incorporates the irreducible component's field of definition and the Culler-Shalen norm.
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Taxonomy
TopicsGeometric and Algebraic Topology · Homotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models