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-1D18A9856A87
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.app.ist.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.app.ist.ac.at/record/6681
M. Filakovský, P. Franek, U. Wagner, S.Y. Zhechev, Journal of Applied and Computational Topology 2 (2018) 177–231.
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>
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. 2018. Computing simplicial representatives of homotopy group elements. Journal of Applied and Computational Topology. 2(3–4), 177–231.
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.
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, 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>
67742019-08-08T06:47:40Z2021-01-12T08:08:58Z