Algorithmic solvability of the lifting-extension problem

TL;DR

This paper presents a polynomial-time algorithm for computing equivariant homotopy classes of maps between finite simplicial sets with group actions, with applications to topological embeddability problems.

Contribution

It introduces the first algorithm for deciding topological embeddability of finite complexes into Euclidean spaces under certain dimensional constraints.

Findings

Algorithm computes all equivariant homotopy classes of maps.

Decides existence of equivariant maps even for higher dimensions.

Applies to topological embeddability of complexes into Euclidean spaces.

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 $d\geq 1$, 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. This yields the first algorithm for deciding topological embeddability of a $k$-dimensional finite simplicial complex into $\mathbb{R}^n$ under the conditions $k\leq\frac 23 n-1$. 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.