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Strong Discrete Morse Theory and Simplicial L–S Category: A Discrete Version of the Lusternik–Schnirelmann Theorem

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Abstract

We prove a discrete version of the Lusternik–Schnirelmann (L–S) theorem for discrete Morse functions and the recently introduced simplicial L–S category of a simplicial complex. To accomplish this, a new notion of critical object of a discrete Morse function is presented, which generalizes the usual concept of critical simplex (in the sense of R. Forman). We show that the non-existence of such critical objects guarantees the strong homotopy equivalence (in the Barmak and Minian’s sense) between the corresponding sublevel complexes. Finally, we establish that the number of critical objects of a discrete Morse function defined on K is an upper bound for the non-normalized simplicial L–S category of K.

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Authors and Affiliations

  1. Departamento de Geometría y Topología, Universidad de Sevilla, Seville, Spain

    Desamparados Fernández-Ternero & José Antonio Vilches

  2. Departamento de Matemáticas, Universidade de Santiago de Compostela, Santiago de Compostela, Spain

    Enrique Macías-Virgós

  3. Department of Mathematics and Computer Science, Ursinus College, 610 E Main Street, Collegeville, PA, 19426, USA

    Nicholas A. Scoville

Authors
  1. Desamparados Fernández-Ternero
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  2. Enrique Macías-Virgós
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  3. Nicholas A. Scoville
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  4. José Antonio Vilches
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Correspondence to Desamparados Fernández-Ternero.

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The first and fourth authors were partially supported by MINECO and FEDER Research Project MTM2015-65397-P and Junta de Andalucía Research Groups FQM-326 and FQM-189. The second author was partially supported by MINECO and FEDER Research Project MTM2016-78647-P.

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Fernández-Ternero, D., Macías-Virgós, E., Scoville, N.A. et al. Strong Discrete Morse Theory and Simplicial L–S Category: A Discrete Version of the Lusternik–Schnirelmann Theorem. Discrete Comput Geom 63, 607–623 (2020). https://doi.org/10.1007/s00454-019-00116-8

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  • DOI: https://doi.org/10.1007/s00454-019-00116-8

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