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Simplicial Lusternik-Schnirelmann category

Desamparados Fernández-Ternero, Enrique Macías-Virgós, Erica Minuz, and José Antonio Vilches

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The simplicial LS-category of a finite abstract simplicial complex is a new invariant of the strong homotopy type, defined in purely combinatorial terms. We prove that it generalizes to arbitrary simplicial complexes the well known notion of arboricity of a graph, and that it allows to develop many notions and results of algebraic topology which are costumary in the classical theory of Lusternik–Schnirelmann category. Also we compare the simplicial category of a complex with the LS-category of its geometric realization and we discuss the simplicial analogue of the Whitehead formulation of the LS-category.

Article information

Publ. Mat., Volume 63, Number 1 (2019), 265-293.

Received: 25 May 2017
Revised: 24 October 2017
First available in Project Euclid: 7 December 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55U10: Simplicial sets and complexes 55M30: Ljusternik-Schnirelman (Lyusternik-Shnirelʹman) category of a space 06F30: Topological lattices, order topologies [See also 06B30, 22A26, 54F05, 54H12]

Lusternik–Schnirelmann category strong homotopy type geometric realization Whitehead formulation of category graph arboricity


Fernández-Ternero, Desamparados; Macías-Virgós, Enrique; Minuz, Erica; Vilches, José Antonio. Simplicial Lusternik-Schnirelmann category. Publ. Mat. 63 (2019), no. 1, 265--293. doi:10.5565/PUBLMAT6311909.

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