---
res:
bibo_abstract:
- "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)
.@eng"
bibo_authorlist:
- foaf_Person:
foaf_givenName: Marek
foaf_name: Filakovský, Marek
foaf_surname: Filakovský
foaf_workInfoHomepage: http://www.librecat.org/personId=3E8AF77E-F248-11E8-B48F-1D18A9856A87
- foaf_Person:
foaf_givenName: Peter
foaf_name: Franek, Peter
foaf_surname: Franek
foaf_workInfoHomepage: http://www.librecat.org/personId=473294AE-F248-11E8-B48F-1D18A9856A87
orcid: 0000-0001-8878-8397
- foaf_Person:
foaf_givenName: Uli
foaf_name: Wagner, Uli
foaf_surname: Wagner
foaf_workInfoHomepage: http://www.librecat.org/personId=36690CA2-F248-11E8-B48F-1D18A9856A87
orcid: 0000-0002-1494-0568
- foaf_Person:
foaf_givenName: Stephan Y
foaf_name: Zhechev, Stephan Y
foaf_surname: Zhechev
foaf_workInfoHomepage: http://www.librecat.org/personId=3AA52972-F248-11E8-B48F-1D18A9856A87
bibo_doi: 10.1007/s41468-018-0021-5
bibo_issue: 3-4
bibo_volume: 2
dct_date: 2018^xs_gYear
dct_isPartOf:
- http://id.crossref.org/issn/2367-1726
- http://id.crossref.org/issn/2367-1734
dct_language: eng
dct_publisher: Springer@
dct_title: Computing simplicial representatives of homotopy group elements@
...