deal with a more general lifting-extension problem. At the same time, lifting-extension problems play a fundamental role in algebraic topology since many problems can be expressed as their instances. We start by explaining our original problem and its concrete applications and then proceed to the main object of our study in this paper – the lifting-extension problem.

Equivariant maps {#equivariant-maps .unnumbered}
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Consider the following algorithmic problem: given a finite group $G$ and two free $G$-spaces $X$ and $Y$, decide the existence of an equivariant map $f{\colon}X\to Y$.

In the particular case $G={{\mathbb Z}}_2$ and $Y=S^{n-1}$ equipped with the antipodal ${{\mathbb Z}}_2$-action, this problem has various applications in geometry and combinatorics.

Concretely, it is well-known that if a simplicial complex $K$ embeds into ${\mathbb R}^{{{n}}}$ then there exists a ${{\mathbb Z}}_2$-equivariant map $(K \times K) \smallsetminus \Delta_K \to S^{n-1}$; the converse holds in the so-called *metastable range* $\dim K \leq \tfrac{2}{3} {{n}}-1$ by [@Weber:PlongementsPolyedresDomaineMetastable-1967]. Algorithmic aspects of the problem of embeddability of $K$ into ${\mathbb R}^{{{n}}}$ were studied in [@MatousekTancerWagner:HardnessEmbeddings-2011] and the meta-stable range was essentially the only remaining case left open. Theorem \[t:emb-metast\] below shows that, for fixed $n$, it is solvable in polynomial time. Theorem \[t:emb-metast\] below shows 