Algorithmic Solvability of the Lifting-Extension Problem

Abstract

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 \(\dim X\le 2d\), for some \(d\ge 1\), we provide an algorithm that computes the set of all equivariant homotopy classes of equivariant continuous maps \(|X|\rightarrow |Y|\); the existence of such a map can be decided even for \(\dim X\le 2d+1\). This yields the first algorithm for deciding topological embeddability of a k-dimensional finite simplicial complex into \(\mathbb R^n\) under the condition \(k\le \frac{2}{3} n-1\). 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.