---
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 $|X|\to|Y|$; the existence of such a map can be decided even for $\dim X\leq 2d+1$. For fixed $G$ and ${{d}}$, the algorithm runs in polynomial time. This yields the first algorithm for deciding topological embeddability of a $k$-dimensional finite simplicial complex into ${\mathbb R}^n$ under the condition $k\leq\frac 23 n-1$.

    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.
author:
- Martin Čadek
- Marek Krčál
- Lukáš Vokřínek
bibliography:
- 'Postnikov.bib'
title: 'Algorithmic solvability of the lifting-extension problem[^1]'
---

Introduction
============

Our original goal for this paper was to design an algorithm that decides existence of an equivariant map between given spaces under a certain “stability” assumption. To explain our solution however, it is more natural to 