Computing simplicial representatives of homotopy group elements

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.