Algorithmic solvability of the lifting-extension problem

Understand

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 $d\geq 1$, 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.