Trisections of 4-manifolds with boundary

Area of Science:

• Topology
• Differential Geometry
• Low-Dimensional Topology

Background:

• Handle decompositions are fundamental tools for studying 4-manifolds.
• Open book decompositions describe the topology of 3-manifolds.
• Trisections offer an alternative to Heegaard splittings for analyzing 4-manifolds.

Purpose of the Study:

• To explicitly construct trisection diagrams from handle decompositions of 4-manifolds with boundary.
• To induce a given open book decomposition on the boundary of the 4-manifold.
• To generalize the existence of relative trisections to 4-manifolds with multiple boundary components.

Main Methods:

• Developing an explicit algorithm based on the handle decomposition of the 4-manifold.
• Relating the construction of trisection diagrams to the existing proof of relative trisection existence.
• Investigating the gluing properties of trisected 4-manifolds with multiple boundary components.

Main Results:

• A concrete procedure for generating trisection diagrams from handle and open book decompositions.
• An extension of the existence result for relative trisections to 4-manifolds with multiple boundary components.
• A method for understanding how trisected 4-manifolds with multiple boundary components can be glued together.

Conclusions:

• The explicit construction provides a practical tool for working with trisections of 4-manifolds.
• The generalization to multiple boundary components broadens the applicability of trisection theory.
• The gluing results are crucial for constructing more complex 4-manifold structures.