Functoriality of group trisections.

Area of Science:

• Topology
• Group Theory
• Category Theory

Background:

• Extends prior work in 3D and 4D topology by Stallings, Jaco, Hempel, Abrams, Kirby, and Gay.
• Focuses on smooth four-dimensional topology and its connections to group theory.

Purpose of the Study:

• To generalize splitting homomorphism and group trisection constructions.
• To establish functors between categories relevant to topology and group theory.
• To deepen the interdisciplinary bridge between topology and group theory.

Main Methods:

• Development of functorial extensions of existing topological constructions.
• Application of category theory to relate topological and algebraic structures.
• Formalization of group trisection and splitting homomorphisms within a categorical framework.

Main Results:

• Successful extension of splitting homomorphism and group trisection constructions to functors.
• Demonstration of a robust connection between smooth 4-manifold topology and the algebraic properties of free and surface groups.
• Establishment of a new framework for studying 4-dimensional topology through the lens of category theory.

Conclusions:

• The functorial approach provides a powerful new tool for understanding smooth 4-dimensional topology.
• This work significantly strengthens the relationship between topology and the theory of free and surface groups.
• The developed framework opens avenues for further research at the intersection of these mathematical fields.