@article{6774,
  abstract     = {A central problem of algebraic topology is to understand the homotopy groups  𝜋𝑑(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  𝜋1(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   𝜋1(𝑋)  trivial), compute the higher homotopy group   𝜋𝑑(𝑋)  for any given   𝑑≥2 . However, these algorithms come with a caveat: They compute the isomorphism type of   𝜋𝑑(𝑋) ,   𝑑≥2  as an abstract finitely generated abelian group given by generators and relations, but they work with very implicit representations of the elements of   𝜋𝑑(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   𝜋𝑑(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   𝑑≥2 , we construct a family of simply connected spaces X such that for any simplicial map representing a generator of   𝜋𝑑(𝑋) , the size of the triangulation of   𝑆𝑑  on which the map is defined, is exponential in size(𝑋) .},
  author       = {Filakovský, 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},
}

@article{1408,
  abstract     = {The concept of well group in a special but important case captures homological properties of the zero set of a continuous map (Formula presented.) on a compact space K that are invariant with respect to perturbations of f. The perturbations are arbitrary continuous maps within (Formula presented.) distance r from f for a given (Formula presented.). The main drawback of the approach is that the computability of well groups was shown only when (Formula presented.) or (Formula presented.). Our contribution to the theory of well groups is twofold: on the one hand we improve on the computability issue, but on the other hand we present a range of examples where the well groups are incomplete invariants, that is, fail to capture certain important robust properties of the zero set. For the first part, we identify a computable subgroup of the well group that is obtained by cap product with the pullback of the orientation of (Formula presented.) by f. In other words, well groups can be algorithmically approximated from below. When f is smooth and (Formula presented.), our approximation of the (Formula presented.)th well group is exact. For the second part, we find examples of maps (Formula presented.) with all well groups isomorphic but whose perturbations have different zero sets. We discuss on a possible replacement of the well groups of vector valued maps by an invariant of a better descriptive power and computability status.},
  author       = {Franek, Peter and Krcál, Marek},
  journal      = {Discrete & Computational Geometry},
  number       = {1},
  pages        = {126 -- 164},
  publisher    = {Springer},
  title        = {{On computability and triviality of well groups}},
  doi          = {10.1007/s00454-016-9794-2},
  volume       = {56},
  year         = {2016},
}