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Computing trisections of 4-manifolds.

Mark Bell1, Joel Hass2, Joachim Hyam Rubinstein3

  • 1Department of Mathematics, University of Illinois, Urbana, IL 61801.

Proceedings of the National Academy of Sciences of the United States of America
|October 24, 2018
PubMed
Summary

We developed an algorithm to compute trisections of four-manifolds from triangulations. This provides complexity bounds for the trisection genus based on the number of 4-simplices used.

Keywords:
computational topologyfour-dimensional triangulationsfour-manifoldtopological algorithmstrisection

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Area of Science:

  • Topology
  • Computational Geometry
  • Low-Dimensional Topology

Background:

  • Four-manifolds are fundamental objects in topology.
  • Trisections offer a powerful decomposition method for four-manifolds.
  • Understanding the complexity of these structures is crucial.

Purpose of the Study:

  • To introduce a novel algorithm for computing trisections of orientable four-manifolds.
  • To establish explicit complexity bounds for the trisection genus.
  • To connect topological properties with combinatorial data from triangulations.

Main Methods:

  • Development of a computational algorithm.
  • Utilizing arbitrary triangulations (collections of 4-simplices or pentachora) as input.
  • Derivation of bounds based on the combinatorial structure of the triangulation.

Main Results:

  • An algorithm to compute trisections of orientable four-manifolds is presented.
  • Explicit complexity bounds for the trisection genus are derived.
  • The bounds are expressed in terms of the number of pentachora in the input triangulation.

Conclusions:

  • The algorithm provides a practical method for analyzing four-manifold trisections.
  • The complexity bounds offer insights into the relationship between combinatorial and topological complexity.
  • This work facilitates further study of four-manifold invariants using computational approaches.