We consider the following decision problem EMBEDk→d in computational topology (where k ≤ d are fixed positive integers): Given a finite simplicial complex K of dimension k, does there exist a (piecewise-linear) embedding of K into ℝd? The special case EMBED1→2 is graph planarity, which is decidable in linear time, as shown by Hopcroft and Tarjan. In higher dimensions, EMBED2→3 and EMBED3→3 are known to be decidable (as well as NP-hard), and recent results of Čadek et al. in computational homotopy theory, in combination with the classical Haefliger–Weber theorem in geometric topology, imply that EMBEDk→d can be solved in polynomial time for any fixed pair (k, d) of dimensions in the so-called metastable range . Here, by contrast, we prove that EMBEDk→d is algorithmically undecidable for almost all pairs of dimensions outside the metastable range, namely for . This almost completely resolves the decidability vs. undecidability of EMBEDk→d in higher dimensions and establishes a sharp dichotomy between polynomial-time solvability and undecidability. Our result complements (and in a wide range of dimensions strengthens) earlier results of Matoušek, Tancer, and the second author, who showed that EMBEDk→d is undecidable for 4 ≤ k ϵ {d – 1, d}, and NP-hard for all remaining pairs (k, d) outside the metastable range and satisfying d ≥ 4.

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(Χ) 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(Χ) 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(Χ), 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(Χ). Converting elements of this abstract group into explicit geometric maps from the d-dimensional sphere Sd to Χ has been an important open question in the emerging field of computational homotopy theory. Here we present an algorithm that, given a simply connected simplicial complex Χ, computes πd(Χ) and represents its elements as simplicial maps from a suitable triangulation of the d-sphere Sd to Χ. For fixed d, the algorithm runs in time exponential in size(X), the number of simplices of Χ. Moreover, we prove that this is optimal: For every fixed d ≥ 2, we construct a family of simply connected simplicial complexes Χ such that for any simplicial map representing a generator of πd(Χ), the size of the triangulation of Sd on which the map is defined is exponential in size(X).