Generalized trisections in all dimensions.

Area of Science:

• Topology
• Geometric Topology
• Manifold Theory

Background:

• Heegaard splittings are fundamental in 3-manifold topology.
• Trisections provide a powerful tool for analyzing 4-manifolds.
• Generalizing these concepts to higher dimensions is a significant challenge.

Purpose of the Study:

• To generalize Heegaard splittings and trisections to arbitrary dimensions.
• To introduce a novel approach using triangulations for manifold decomposition.
• To explore the structure of resulting handlebodies and their intersections.

Main Methods:

• Utilizing triangulations as a primary tool for manifold decomposition.
• Developing a method to divide closed piecewise linear n-manifolds into n-dimensional 1-handlebodies.
• Analyzing the dimensionality of the spines of intersections between handlebodies.

Main Results:

• Demonstrated that any closed piecewise linear n-manifold can be decomposed into a specific number of n-dimensional 1-handlebodies.
• Established that the intersections of these handlebodies possess spines of small dimensions.
• Provided several applications and generalizations of the developed approach.

Conclusions:

• The triangulation-based method offers a unified framework for understanding manifold decompositions across all dimensions.
• This generalization provides new insights into the structure of higher-dimensional manifolds.
• The approach facilitates novel constructions and further theoretical developments in topology.