Algorithmic Solvability of the Lifting-Extension Problem

Let X and Y be finite simplicial sets (e.g. finite simplicial complexes), both equipped with a free simplicial action of a finite group G. Assuming that Y is d-connected and $$\dim X\le 2d$$dimX≤2d, for some $$d\ge 1$$d≥1, we provide an algorithm that computes the set of all equivariant homotopy classes of equivariant continuous maps $$|X|\rightarrow |Y|$$|X|→|Y|; the existence of such a map can be decided even for $$\dim X\le 2d+1$$dimX≤2d+1. This yields the first algorithm for deciding topological embeddability of a k-dimensional finite simplicial complex into $$\mathbb R^n$$Rn under the condition $$k\le \frac{2}{3} n-1$$k≤23n-1. More generally, we present an algorithm that, given a lifting-extension problem satisfying an appropriate stability assumption, computes the set of all homotopy classes of solutions. This result is new even in the non-equivariant situation.