Computing simplicial representatives of homotopy group elements
Marek Filakovský, Peter Franek, Uli Wagner, Stephan Zhechev · Journal of Applied and Computational Topology · 2018
A central problem of algebraic topology is to understand the homotopy groups $$\pi _d(X)$$ 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 $$\pi _1(X)$$ 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 $$\pi _1(X)$$ trivial), compute the higher homotopy group $$\pi _d(X)$$ for any given $$d\ge 2$$ . However, these algorithms come with a caveat: They compute the isomorphism type of $$\pi _d(X)$$ , $$d\ge 2$$ as an abstract finitely generated abelian group given by generators and relations, but they work with very implicit representations of the elements of $$\pi _d(X)$$ . Converting elements of this abstract group into explicit geometric maps from the d-dimensional sphere $$S^d$$ 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 $$\pi _d(X)$$ and represents its elements as simplicial maps from a suitable triangulation of the d-sphere $$S^d$$ to X. For fixed d, the algorithm runs in time exponential in $$\mathrm {size}(X)$$ , the number of simplices of X. Moreover, we prove that this is optimal: For every fixed $$d\ge 2$$ , we construct a family of simply connected spaces X such that for any simplicial map representing a generator of $$\pi _d(X)$$ , the size of the triangulation of $$S^d$$ on which the map is defined, is exponential in $$\mathrm {size}(X)$$ .