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. We provide an algorithm that computes the set of all equivariant homotopy classes of equivariant continuous maps.

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.

Keywords

• Homotopy classes
• Equivariant
• Fibrewise
• Lifting-extension problem
• Algorithmic computation
• Embeddability
• Moore-Postnikov tower