article
Computing simplicial representatives of homotopy group elements
published
yes
Marek
Filakovský
author 3E8AF77E-F248-11E8-B48F-1D18A9856A87
Peter
Franek
author 473294AE-F248-11E8-B48F-1D18A9856A870000-0001-8878-8397
Uli
Wagner
author 36690CA2-F248-11E8-B48F-1D18A9856A870000-0002-1494-0568
Stephan Y
Zhechev
author 3AA52972-F248-11E8-B48F-1D18A9856A87
UlWa
department
Robust invariants of Nonlinear Systems
project
FWF Open Access Fund
project
A central problem of algebraic topology is to understand the homotopy groups 𝜋𝑑(𝑋) of a topological space X. For the computational version of the problem, it is well known that there is no algorithm to decide whether the fundamental group 𝜋1(𝑋) of a given finite simplicial complex X is trivial. On the other hand, there are several algorithms that, given a finite simplicial complex X that is simply connected (i.e., with 𝜋1(𝑋) trivial), compute the higher homotopy group 𝜋𝑑(𝑋) for any given 𝑑≥2 . However, these algorithms come with a caveat: They compute the isomorphism type of 𝜋𝑑(𝑋) , 𝑑≥2 as an abstract finitely generated abelian group given by generators and relations, but they work with very implicit representations of the elements of 𝜋𝑑(𝑋) . Converting elements of this abstract group into explicit geometric maps from the d-dimensional sphere 𝑆𝑑 to X has been one of the main unsolved problems in the emerging field of computational homotopy theory. Here we present an algorithm that, given a simply connected space X, computes 𝜋𝑑(𝑋) and represents its elements as simplicial maps from a suitable triangulation of the d-sphere 𝑆𝑑 to X. For fixed d, the algorithm runs in time exponential in size(𝑋) , the number of simplices of X. Moreover, we prove that this is optimal: For every fixed 𝑑≥2 , we construct a family of simply connected spaces X such that for any simplicial map representing a generator of 𝜋𝑑(𝑋) , the size of the triangulation of 𝑆𝑑 on which the map is defined, is exponential in size(𝑋) .
https://research-explorer.ista.ac.at/download/6774/6775/2018_JourAppliedComputTopology_Filakovsky.pdf
application/pdfno
https://creativecommons.org/licenses/by/4.0/
Springer2018
eng
Journal of Applied and Computational Topology
2367-1726
2367-173410.1007/s41468-018-0021-5
23-4177-231
https://research-explorer.ista.ac.at/record/6681
Filakovský, Marek, Peter Franek, Uli Wagner, and Stephan Y Zhechev. “Computing Simplicial Representatives of Homotopy Group Elements.” <i>Journal of Applied and Computational Topology</i>. Springer, 2018. <a href="https://doi.org/10.1007/s41468-018-0021-5">https://doi.org/10.1007/s41468-018-0021-5</a>.
Filakovský M, Franek P, Wagner U, Zhechev SY. 2018. Computing simplicial representatives of homotopy group elements. Journal of Applied and Computational Topology. 2(3–4), 177–231.
Filakovský, M., Franek, P., Wagner, U., & Zhechev, S. Y. (2018). Computing simplicial representatives of homotopy group elements. <i>Journal of Applied and Computational Topology</i>. Springer. <a href="https://doi.org/10.1007/s41468-018-0021-5">https://doi.org/10.1007/s41468-018-0021-5</a>
M. Filakovský, P. Franek, U. Wagner, S.Y. Zhechev, Journal of Applied and Computational Topology 2 (2018) 177–231.
Filakovský, Marek, et al. “Computing Simplicial Representatives of Homotopy Group Elements.” <i>Journal of Applied and Computational Topology</i>, vol. 2, no. 3–4, Springer, 2018, pp. 177–231, doi:<a href="https://doi.org/10.1007/s41468-018-0021-5">10.1007/s41468-018-0021-5</a>.
Filakovský M, Franek P, Wagner U, Zhechev SY. Computing simplicial representatives of homotopy group elements. <i>Journal of Applied and Computational Topology</i>. 2018;2(3-4):177-231. doi:<a href="https://doi.org/10.1007/s41468-018-0021-5">10.1007/s41468-018-0021-5</a>
M. Filakovský, P. Franek, U. Wagner, and S. Y. Zhechev, “Computing simplicial representatives of homotopy group elements,” <i>Journal of Applied and Computational Topology</i>, vol. 2, no. 3–4. Springer, pp. 177–231, 2018.
67742019-08-08T06:47:40Z2024-10-09T20:58:57Z