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Computing Homotopy Classes for Diagrams.

Marek Filakovský1, Lukáš Vokřínek2

  • 1Department of Algebra, Charles University, Sokolovská 49/83, 186 75 Prague 8, Czech Republic.

Discrete & Computational Geometry
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Summary
This summary is machine-generated.

This study introduces algorithms for computing homotopy classes of maps between simplicial set diagrams, crucial for Tverberg-type problems in computational topology. The algorithms offer polynomial-time solutions under stability conditions, enhancing computational topology research.

Keywords:
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Area of Science:

  • Algebraic Topology
  • Computational Topology
  • Discrete Geometry

Background:

  • The problem of computing homotopy classes of maps between diagrams of simplicial sets is fundamental in algebraic topology.
  • Decidability issues arise when stability conditions are relaxed, posing challenges for algorithmic solutions.
  • Elmendorf's theorem provides a bridge between group actions on simplicial sets and their representation theory.

Purpose of the Study:

  • To develop an algorithm for computing homotopy classes of maps between diagrams of simplicial sets.
  • To extend this algorithm to compute homotopy classes of equivariant maps under group actions.
  • To apply these algorithmic advancements to solve Tverberg-type problems in computational topology.

Main Methods:

  • The core method involves constructing an algorithm that computes the set of homotopy classes of maps of diagrams.
  • Elmendorf's theorem is utilized to derive an algorithm for equivariant maps, incorporating group actions.
  • The stability condition is crucial for ensuring polynomial-time computability.

Main Results:

  • An algorithm is presented that computes homotopy classes of maps of simplicial set diagrams in polynomial time for fixed parameters.
  • A novel algorithm is deduced for computing homotopy classes of equivariant maps, leveraging Elmendorf's theorem.
  • The Tverberg-type problem concerning maps without r-tuple intersection points is shown to be algorithmically decidable in polynomial time for fixed dimensions and parameters.

Conclusions:

  • The developed algorithms provide efficient computational tools for problems in algebraic and computational topology.
  • The work demonstrates the power of combining abstract algebraic concepts with algorithmic approaches.
  • This research contributes to the decidability and algorithmic tractability of geometric intersection problems.