@article{6774,
abstract = {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) .},
author = {FilakovskΓ½, Marek and Franek, Peter and Wagner, Uli and Zhechev, Stephan Y},
issn = {2367-1734},
journal = {Journal of Applied and Computational Topology},
number = {3-4},
pages = {177--231},
publisher = {Springer},
title = {{Computing simplicial representatives of homotopy group elements}},
doi = {10.1007/s41468-018-0021-5},
volume = {2},
year = {2018},
}