Algorithmic Solvability of the Lifting Extension Problem Journal Article


Author(s): Čadek, Martin; Krčál, Marek; Vokřínek, Lukáš
Article Title: Algorithmic Solvability of the Lifting Extension Problem
Alternate Title: Discrete and Computational Geometry
Affiliation IST Austria
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 dimX≤2d, 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 dimX≤2d+1. This yields the first algorithm for deciding topological embeddability of a k-dimensional finite simplicial complex into Rn under the condition k≤23n−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.
Keywords: Homotopy classes; equivariant; fibrewise; lifting-extension problem; algorithmic computation; Embeddability; Moore–Postnikov tower
Journal Title: Discrete & Computational Geometry
Volume: 54
Issue 4
ISSN: 0179-5376
Publisher: Springer  
Date Published: 2017-06-01
Start Page: 915
End Page: 965
DOI: 10.1007/s00454-016-9855-6
Notes: The research of M. Č. was supported by the Project CZ.1.07/2.3.00/20.0003 of the Operational Programme Education for Competitiveness of the Ministry of Education, Youth and Sports of the Czech Republic. The research by M. K. was supported by the Center of Excellence—Inst. for Theor. Comput. Sci., Prague (Project P202/12/G061 of GA ČR) and by the Project LL1201 ERCCZ CORES. The research of L. V. was supported by the Center of Excellence—Eduard Čech Institute (Project P201/12/G028 of GA ČR).
Open access: no
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