Computing simplicial representatives of homotopy group elements
FilakovskΓ½, Marek
Franek, Peter
Wagner, Uli
Zhechev, Stephan Y
ddc:514
A central problem of algebraic topology is to understand the homotopy groups ππ(π) 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(π) 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(π) trivial), compute the higher homotopy group ππ(π) for any given πβ₯2 . However, these algorithms come with a caveat: They compute the isomorphism type of ππ(π) , πβ₯2 as an abstract finitely generated abelian group given by generators and relations, but they work with very implicit representations of the elements of ππ(π) . Converting elements of this abstract group into explicit geometric maps from the d-dimensional sphere ππ 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 ππ(π) and represents its elements as simplicial maps from a suitable triangulation of the d-sphere ππ to X. For fixed d, the algorithm runs in time exponential in size(π) , the number of simplices of X. Moreover, we prove that this is optimal: For every fixed πβ₯2 , we construct a family of simply connected spaces X such that for any simplicial map representing a generator of ππ(π) , the size of the triangulation of ππ on which the map is defined, is exponential in size(π) .
Springer
2018
info:eu-repo/semantics/article
doc-type:article
text
http://purl.org/coar/resource_type/c_6501
https://research-explorer.ista.ac.at/record/6774
https://research-explorer.ista.ac.at/download/6774/6775
FilakovskΓ½ 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>
eng
info:eu-repo/semantics/altIdentifier/doi/10.1007/s41468-018-0021-5
info:eu-repo/semantics/altIdentifier/issn/2367-1726
info:eu-repo/semantics/altIdentifier/issn/2367-1734
info:eu-repo/grantAgreement/FWF//M01980
info:eu-repo/semantics/openAccess