Abstract
We undertake a systematic study of fibrations in the setting of abstract simplicial complexes, where the concept of “homotopy” has been replaced by that of “contiguity”. Then, a fibration will be a simplicial map satisfying the “contiguity lifting property”. This definition turns out to be equivalent to that introduced by Minian, established in terms of a cylinder construction \(K \times I_m\). This allows us to prove several properties of simplicial fibrations which are analogous to the classical ones in the topological setting, for instance: all the fibers of a fibration with connected base have the same strong homotopy type and any fibration with a strongly collapsible base is fibrewise trivial. We also introduce the concept of “simplicial finite-fibration”, that is, a simplicial map which has the contiguity lifting property only for finite complexes. Then, we prove that the path fibration \(\mathrm {P}K \rightarrow K\times K\) is a finite-fibration, where \(\mathrm {P}K\) is the simplicial complex of Moore paths introduced by Grandis. This result allows us to prove that any simplicial map factors through a finite-fibration, up to a P-homotopy equivalence. Moreover, we prove a simplicial version of a Varadarajan result for fibrations, relating the LS-category of the total space, the base and the generic fiber. Finally, we introduce a definition of “Švarc genus” of a simplicial map and we are able to compare the Švarc genus of path fibrations with the notions of simplicial LS-category and simplicial topological complexity introduced by the authors in several previous papers.
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The first and fourth authors were partially supported by MINECO Spain Research Project MTM2015–65397–P and Junta de Andalucía Research Groups FQM–326 and FQM–189. The second and third authors were partially supported by MINECO-FEDER research project MTM2016–78647–P.
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Fernández-Ternero, D., García-Calcines, J.M., Macías-Virgós, E. et al. Simplicial fibrations. RACSAM 115, 54 (2021). https://doi.org/10.1007/s13398-020-00966-5
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DOI: https://doi.org/10.1007/s13398-020-00966-5