Algorithmic solvability of the lifting-extension problem

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 h...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2016-10
Main Authors: Čadek, Martin, Krčál, Marek, Vokřínek, Lukáš
Format: Article
Language:English
Subjects:
Algorithms
Polynomials
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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.
ISSN:2331-8422