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 ≤ 2 d , for some d ≥ 1 , we provide an algorithm that computes the set of all equivariant homotopy classes of equivari...

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Bibliographic Details
Published in:Discrete & computational geometry 2017-06, Vol.57 (4), p.915-965
Main Authors: Cadek, Martin, KrAeal, Marek, Vokrinek, Lukas
Format: Article
Language:English
Subjects:
Algorithms
Combinatorics
Computational geometry
Computational Mathematics and Numerical Analysis
Mathematical analysis
Mathematical models
Mathematical problems
Mathematics
Mathematics and Statistics
Stability
Texts
Topology
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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 ≤ 2 d , for some d ≥ 1 , we provide an algorithm that computes the set of all equivariant homotopy classes of equivariant continuous maps | X | → | Y | ; the existence of such a map can be decided even for dim X ≤ 2 d + 1 . This yields the first algorithm for deciding topological embeddability of a k -dimensional finite simplicial complex into R n under the condition k ≤ 2 3 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:0179-5376
1432-0444
DOI:10.1007/s00454-016-9855-6