Computing simplicial representatives of homotopy group elements

M. FilakovskΓ½, P. Franek, U. Wagner, S.Y. Zhechev, Journal of Applied and Computational Topology 2 (2018) 177–231.

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Journal Article | Published | English
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Abstract
A central problem of algebraic topology is to understand the homotopy groups πœ‹π‘‘(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 πœ‹1(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 πœ‹1(𝑋) trivial), compute the higher homotopy group πœ‹π‘‘(𝑋) for any given 𝑑β‰₯2 . However, these algorithms come with a caveat: They compute the isomorphism type of πœ‹π‘‘(𝑋) , 𝑑β‰₯2 as an abstract finitely generated abelian group given by generators and relations, but they work with very implicit representations of the elements of πœ‹π‘‘(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 πœ‹π‘‘(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 𝑑β‰₯2 , we construct a family of simply connected spaces X such that for any simplicial map representing a generator of πœ‹π‘‘(𝑋) , the size of the triangulation of 𝑆𝑑 on which the map is defined, is exponential in size(𝑋) .
Publishing Year
Date Published
2018-12-01
Journal Title
Journal of Applied and Computational Topology
Volume
2
Issue
3-4
Page
177-231
ISSN
eISSN
IST-REx-ID

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FilakovskΓ½ M, Franek P, Wagner U, Zhechev SY. Computing simplicial representatives of homotopy group elements. Journal of Applied and Computational Topology. 2018;2(3-4):177-231. doi:10.1007/s41468-018-0021-5
FilakovskΓ½, M., Franek, P., Wagner, U., & Zhechev, S. Y. (2018). Computing simplicial representatives of homotopy group elements. Journal of Applied and Computational Topology, 2(3–4), 177–231. https://doi.org/10.1007/s41468-018-0021-5
FilakovskΓ½, Marek, Peter Franek, Uli Wagner, and Stephan Y Zhechev. β€œComputing Simplicial Representatives of Homotopy Group Elements.” Journal of Applied and Computational Topology 2, no. 3–4 (2018): 177–231. https://doi.org/10.1007/s41468-018-0021-5.
M. FilakovskΓ½, P. Franek, U. Wagner, and S. Y. Zhechev, β€œComputing simplicial representatives of homotopy group elements,” Journal of Applied and Computational Topology, vol. 2, no. 3–4, pp. 177–231, 2018.
FilakovskΓ½ 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.
FilakovskΓ½, Marek, et al. β€œComputing Simplicial Representatives of Homotopy Group Elements.” Journal of Applied and Computational Topology, vol. 2, no. 3–4, Springer, 2018, pp. 177–231, doi:10.1007/s41468-018-0021-5.
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