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Summary
This summary is machine-generated.

This study presents an algorithm to compute higher homotopy groups for simply connected spaces. It represents group elements as explicit geometric maps, solving a key problem in computational homotopy theory.

Keywords:
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Area of Science:

  • Algebraic topology
  • Computational homotopy theory
  • Computational topology

Background:

  • Understanding homotopy groups is central to algebraic topology.
  • Algorithms exist to compute higher homotopy groups for simply connected spaces.
  • Existing methods represent groups abstractly, lacking explicit geometric element representation.

Purpose of the Study:

  • To develop an algorithm for computing higher homotopy groups.
  • To represent elements of homotopy groups as explicit simplicial maps.
  • To address the challenge of converting abstract group representations into geometric maps.

Main Methods:

  • Developed an algorithm to compute higher homotopy groups for simply connected spaces.
  • Represented elements of homotopy groups as simplicial maps from spheres to the space.
  • Analyzed algorithm runtime and optimality for fixed dimensions.

Main Results:

  • An algorithm computes higher homotopy groups and represents elements as simplicial maps.
  • For fixed dimensions, the algorithm's runtime is exponential in the input space's complexity.
  • Proved optimality by constructing spaces requiring exponentially sized triangulations for map representation.

Conclusions:

  • The presented algorithm provides explicit geometric representations for homotopy group elements.
  • The computational complexity is shown to be optimally exponential for fixed dimensions.
  • This work advances computational homotopy theory by bridging abstract and geometric group representations.