---
_id: '7806'
abstract:
- lang: eng
text: "We consider the following decision problem EMBEDk→d in computational topology
(where k ≤ d are fixed positive integers): Given a finite simplicial complex K
of dimension k, does there exist a (piecewise-linear) embedding of K into ℝd?\r\nThe
special case EMBED1→2 is graph planarity, which is decidable in linear time, as
shown by Hopcroft and Tarjan. In higher dimensions, EMBED2→3 and EMBED3→3 are
known to be decidable (as well as NP-hard), and recent results of Čadek et al.
in computational homotopy theory, in combination with the classical Haefliger–Weber
theorem in geometric topology, imply that EMBEDk→d can be solved in polynomial
time for any fixed pair (k, d) of dimensions in the so-called metastable range
.\r\nHere, by contrast, we prove that EMBEDk→d is algorithmically undecidable
for almost all pairs of dimensions outside the metastable range, namely for .
This almost completely resolves the decidability vs. undecidability of EMBEDk→d
in higher dimensions and establishes a sharp dichotomy between polynomial-time
solvability and undecidability.\r\nOur result complements (and in a wide range
of dimensions strengthens) earlier results of Matoušek, Tancer, and the second
author, who showed that EMBEDk→d is undecidable for 4 ≤ k ϵ {d – 1, d}, and NP-hard
for all remaining pairs (k, d) outside the metastable range and satisfying d ≥
4."
article_processing_charge: No
author:
- first_name: Marek
full_name: Filakovský, Marek
id: 3E8AF77E-F248-11E8-B48F-1D18A9856A87
last_name: Filakovský
- first_name: Uli
full_name: Wagner, Uli
id: 36690CA2-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
orcid: 0000-0002-1494-0568
- first_name: Stephan Y
full_name: Zhechev, Stephan Y
id: 3AA52972-F248-11E8-B48F-1D18A9856A87
last_name: Zhechev
citation:
ama: 'Filakovský M, Wagner U, Zhechev SY. Embeddability of simplicial complexes
is undecidable. In: Proceedings of the Annual ACM-SIAM Symposium on Discrete
Algorithms. Vol 2020-January. SIAM; 2020:767-785. doi:10.1137/1.9781611975994.47'
apa: 'Filakovský, M., Wagner, U., & Zhechev, S. Y. (2020). Embeddability of
simplicial complexes is undecidable. In Proceedings of the Annual ACM-SIAM
Symposium on Discrete Algorithms (Vol. 2020–January, pp. 767–785). Salt Lake
City, UT, United States: SIAM. https://doi.org/10.1137/1.9781611975994.47'
chicago: Filakovský, Marek, Uli Wagner, and Stephan Y Zhechev. “Embeddability of
Simplicial Complexes Is Undecidable.” In Proceedings of the Annual ACM-SIAM
Symposium on Discrete Algorithms, 2020–January:767–85. SIAM, 2020. https://doi.org/10.1137/1.9781611975994.47.
ieee: M. Filakovský, U. Wagner, and S. Y. Zhechev, “Embeddability of simplicial
complexes is undecidable,” in Proceedings of the Annual ACM-SIAM Symposium
on Discrete Algorithms, Salt Lake City, UT, United States, 2020, vol. 2020–January,
pp. 767–785.
ista: 'Filakovský M, Wagner U, Zhechev SY. 2020. Embeddability of simplicial complexes
is undecidable. Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms.
SODA: Symposium on Discrete Algorithms vol. 2020–January, 767–785.'
mla: Filakovský, Marek, et al. “Embeddability of Simplicial Complexes Is Undecidable.”
Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms, vol.
2020–January, SIAM, 2020, pp. 767–85, doi:10.1137/1.9781611975994.47.
short: M. Filakovský, U. Wagner, S.Y. Zhechev, in:, Proceedings of the Annual ACM-SIAM
Symposium on Discrete Algorithms, SIAM, 2020, pp. 767–785.
conference:
end_date: 2020-01-08
location: Salt Lake City, UT, United States
name: 'SODA: Symposium on Discrete Algorithms'
start_date: 2020-01-05
date_created: 2020-05-10T22:00:48Z
date_published: 2020-01-01T00:00:00Z
date_updated: 2021-01-12T08:15:38Z
day: '01'
department:
- _id: UlWa
doi: 10.1137/1.9781611975994.47
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.1137/1.9781611975994.47
month: '01'
oa: 1
oa_version: Published Version
page: 767-785
project:
- _id: 26611F5C-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: P31312
name: Algorithms for Embeddings and Homotopy Theory
publication: Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms
publication_identifier:
isbn:
- '9781611975994'
publication_status: published
publisher: SIAM
quality_controlled: '1'
scopus_import: 1
status: public
title: Embeddability of simplicial complexes is undecidable
type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 2020-January
year: '2020'
...
---
_id: '6563'
abstract:
- lang: eng
text: "This paper presents two algorithms. The first decides the existence of a
pointed homotopy between given simplicial maps \U0001D453,\U0001D454:\U0001D44B→\U0001D44C,
and the second computes the group [\U0001D6F4\U0001D44B,\U0001D44C]∗ of pointed
homotopy classes of maps from a suspension; in both cases, the target Y is assumed
simply connected. More generally, these algorithms work relative to \U0001D434⊆\U0001D44B."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Marek
full_name: Filakovský, Marek
id: 3E8AF77E-F248-11E8-B48F-1D18A9856A87
last_name: Filakovský
- first_name: Lukas
full_name: Vokřínek, Lukas
last_name: Vokřínek
citation:
ama: Filakovský M, Vokřínek L. Are two given maps homotopic? An algorithmic viewpoint.
Foundations of Computational Mathematics. 2020;20:311-330. doi:10.1007/s10208-019-09419-x
apa: Filakovský, M., & Vokřínek, L. (2020). Are two given maps homotopic? An
algorithmic viewpoint. Foundations of Computational Mathematics. Springer
Nature. https://doi.org/10.1007/s10208-019-09419-x
chicago: Filakovský, Marek, and Lukas Vokřínek. “Are Two given Maps Homotopic? An
Algorithmic Viewpoint.” Foundations of Computational Mathematics. Springer
Nature, 2020. https://doi.org/10.1007/s10208-019-09419-x.
ieee: M. Filakovský and L. Vokřínek, “Are two given maps homotopic? An algorithmic
viewpoint,” Foundations of Computational Mathematics, vol. 20. Springer
Nature, pp. 311–330, 2020.
ista: Filakovský M, Vokřínek L. 2020. Are two given maps homotopic? An algorithmic
viewpoint. Foundations of Computational Mathematics. 20, 311–330.
mla: Filakovský, Marek, and Lukas Vokřínek. “Are Two given Maps Homotopic? An Algorithmic
Viewpoint.” Foundations of Computational Mathematics, vol. 20, Springer
Nature, 2020, pp. 311–30, doi:10.1007/s10208-019-09419-x.
short: M. Filakovský, L. Vokřínek, Foundations of Computational Mathematics 20 (2020)
311–330.
date_created: 2019-06-16T21:59:14Z
date_published: 2020-04-01T00:00:00Z
date_updated: 2023-08-17T13:50:44Z
day: '01'
department:
- _id: UlWa
doi: 10.1007/s10208-019-09419-x
external_id:
arxiv:
- '1312.2337'
isi:
- '000522437400004'
intvolume: ' 20'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1312.2337
month: '04'
oa: 1
oa_version: Preprint
page: 311-330
project:
- _id: 26611F5C-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: P31312
name: Algorithms for Embeddings and Homotopy Theory
publication: Foundations of Computational Mathematics
publication_identifier:
eissn:
- '16153383'
issn:
- '16153375'
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Are two given maps homotopic? An algorithmic viewpoint
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 20
year: '2020'
...
---
_id: '6774'
abstract:
- lang: eng
text: "A central problem of algebraic topology is to understand the homotopy groups
\ \U0001D70B\U0001D451(\U0001D44B) 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 \U0001D70B1(\U0001D44B) 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 \U0001D70B1(\U0001D44B)
\ trivial), compute the higher homotopy group \U0001D70B\U0001D451(\U0001D44B)
\ for any given \U0001D451≥2 . However, these algorithms come with a caveat:
They compute the isomorphism type of \U0001D70B\U0001D451(\U0001D44B) , \U0001D451≥2
\ as an abstract finitely generated abelian group given by generators and relations,
but they work with very implicit representations of the elements of \U0001D70B\U0001D451(\U0001D44B)
. Converting elements of this abstract group into explicit geometric maps from
the d-dimensional sphere \U0001D446\U0001D451 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 \U0001D70B\U0001D451(\U0001D44B)
\ and represents its elements as simplicial maps from a suitable triangulation
of the d-sphere \U0001D446\U0001D451 to X. For fixed d, the algorithm runs
in time exponential in size(\U0001D44B) , the number of simplices of X. Moreover,
we prove that this is optimal: For every fixed \U0001D451≥2 , we construct a
family of simply connected spaces X such that for any simplicial map representing
a generator of \U0001D70B\U0001D451(\U0001D44B) , the size of the triangulation
of \U0001D446\U0001D451 on which the map is defined, is exponential in size(\U0001D44B)
."
article_type: original
author:
- first_name: Marek
full_name: Filakovský, Marek
id: 3E8AF77E-F248-11E8-B48F-1D18A9856A87
last_name: Filakovský
- first_name: Peter
full_name: Franek, Peter
id: 473294AE-F248-11E8-B48F-1D18A9856A87
last_name: Franek
orcid: 0000-0001-8878-8397
- first_name: Uli
full_name: Wagner, Uli
id: 36690CA2-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
orcid: 0000-0002-1494-0568
- first_name: Stephan Y
full_name: Zhechev, Stephan Y
id: 3AA52972-F248-11E8-B48F-1D18A9856A87
last_name: Zhechev
citation:
ama: Filakovský 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
apa: Filakovský, M., Franek, P., Wagner, U., & Zhechev, S. Y. (2018). Computing
simplicial representatives of homotopy group elements. Journal of Applied and
Computational Topology. Springer. https://doi.org/10.1007/s41468-018-0021-5
chicago: Filakovský, Marek, Peter Franek, Uli Wagner, and Stephan Y Zhechev. “Computing
Simplicial Representatives of Homotopy Group Elements.” Journal of Applied
and Computational Topology. Springer, 2018. https://doi.org/10.1007/s41468-018-0021-5.
ieee: M. Filakovský, P. Franek, U. Wagner, and S. Y. Zhechev, “Computing simplicial
representatives of homotopy group elements,” Journal of Applied and Computational
Topology, vol. 2, no. 3–4. Springer, pp. 177–231, 2018.
ista: 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.
mla: Filakovský, Marek, et al. “Computing 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.
short: M. Filakovský, P. Franek, U. Wagner, S.Y. Zhechev, Journal of Applied and
Computational Topology 2 (2018) 177–231.
date_created: 2019-08-08T06:47:40Z
date_published: 2018-12-01T00:00:00Z
date_updated: 2023-09-07T13:10:36Z
day: '01'
ddc:
- '514'
department:
- _id: UlWa
doi: 10.1007/s41468-018-0021-5
file:
- access_level: open_access
checksum: cf9e7fcd2a113dd4828774fc75cdb7e8
content_type: application/pdf
creator: dernst
date_created: 2019-08-08T06:55:21Z
date_updated: 2020-07-14T12:47:40Z
file_id: '6775'
file_name: 2018_JourAppliedComputTopology_Filakovsky.pdf
file_size: 1056278
relation: main_file
file_date_updated: 2020-07-14T12:47:40Z
has_accepted_license: '1'
intvolume: ' 2'
issue: 3-4
language:
- iso: eng
month: '12'
oa: 1
oa_version: Published Version
page: 177-231
project:
- _id: 25F8B9BC-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: M01980
name: Robust invariants of Nonlinear Systems
- _id: 3AC91DDA-15DF-11EA-824D-93A3E7B544D1
call_identifier: FWF
name: FWF Open Access Fund
publication: Journal of Applied and Computational Topology
publication_identifier:
eissn:
- 2367-1734
issn:
- 2367-1726
publication_status: published
publisher: Springer
quality_controlled: '1'
related_material:
record:
- id: '6681'
relation: dissertation_contains
status: public
status: public
title: Computing simplicial representatives of homotopy group elements
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 2
year: '2018'
...