Illinois Journal of Mathematics

Lusternik–Schnirelmann category for simplicial complexes

Seth Aaronson and Nicholas A. Scoville

Full-text: Open access

Abstract

The discrete version of Morse theory due to Robin Forman is a powerful tool utilized in the study of topology, combinatorics, and mathematics involving the overlap of these fields. Inspired by the success of discrete Morse theory, we take the first steps in defining a discrete version of the Lusternik–Schnirelmann category suitable for simplicial complexes. This invariant is based on collapsibility as opposed to contractibility, and is defined in the spirit of the geometric category of a topological space. We prove some basic results of this theory, showing where it agrees and differs from that of the smooth case. Our work culminates in a discrete version of the Lusternik–Schnirelmann theorem relating the number of critical points of a discrete Morse function to its discrete category.

Article information

Source
Illinois J. Math., Volume 57, Number 3 (2013), 743-753.

Dates
First available in Project Euclid: 3 November 2014

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1415023508

Mathematical Reviews number (MathSciNet)
MR3275736

Zentralblatt MATH identifier
1302.55004

Subjects
Primary: 55M30: Ljusternik-Schnirelman (Lyusternik-Shnirelʹman) category of a space 55U05: Abstract complexes
Secondary: 57M15: Relations with graph theory [See also 05Cxx]

Citation

Aaronson, Seth; Scoville, Nicholas A. Lusternik–Schnirelmann category for simplicial complexes. Illinois J. Math. 57 (2013), no. 3, 743--753. https://projecteuclid.org/euclid.ijm/1415023508


Export citation

References

  • R. H. Bing, Some aspects of the topology of $3$-manifolds related to the Poincaré conjecture, Lectures on Modern Mathematics, vol. II, Wiley, New York, 1964, pp. 93–128.
  • M. M. Cohen, A course in simple-homotopy theory, Graduate Texts in Mathematics, vol. 10, Springer, New York, 1973.
  • O. Cornea, G. Lupton, J. Oprea and D. Tanré, Lusternik–Schnirelmann category, Mathematical Surveys and Monographs, vol. 103, Amer. Math. Soc., Providence, RI, 2003.
  • D. L. Ferrario and R. A. Piccinini, Simplicial structures in topology, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer, New York, 2011. Translated from the 2009 Italian original by Maria Nair Piccinini.
  • R. Forman, Morse theory for cell complexes, Adv. Math. 134 (1998), no. 1, 90–145.
  • R. Forman, Morse theory and evasiveness, Combinatorica 20 (2000), no. 4, 489–504.
  • R. Forman, A user's guide to discrete Morse theory, Sém. Lothar. Combin. 48 (2002), \bnumberArt. B48c, 1–35.
  • R. Forman, Some applications of combinatorial differential topology, Graphs and patterns in mathematics and theoretical physics, Proc. Sympos. Pure Math., vol. 73, Amer. Math. Soc., Providence, RI, 2005, pp. 281–313.
  • R. H. Fox, On the Lusternik–Schnirelmann category, Ann. of Math. (2) 42 (1941), 333–370.
  • R. Fritsch and R. A. Piccinini, Cellular structures in topology, Cambridge Studies in Advanced Mathematics, vol. 19, Cambridge Univ. Press, Cambridge, 1990.
  • F. Harary, Graph theory, Addison-Wesley Publishing Co., Reading, MA–Menlo Park, CA–London, 1969.
  • D. Kozlov, Combinatorial algebraic topology, Algorithms and Computation in Mathematics, vol. 21, Springer, Berlin, 2008.
  • I. Lazăr, Applications to discrete Morse theory: The collapsibility of CAT(0) cubical complexes of dimension 2 and 3, Carpathian J. Math. 27 (2011), no. 2, 225–237.
  • L. Lusternik and L. Schnirelmann, Méthodes topologiques dans les problémes variationnels, Hermann, Paris, 1934.
  • W. S. Massey, A basic course in algebraic topology, Graduate Texts in Mathematics, vol. 127, Springer, New York, 1991.
  • F. Mori and M. Salvetti, (Discrete) Morse theory on configuration spaces, Math. Res. Lett. 18 (2011), no. 1, 39–57.
  • C. S. J. A. Nash-Williams, Edge-disjoint spanning trees of finite graphs, J. Lond. Math. Soc. (2) 36 (1961), 445–450.
  • J. J. Rotman, An introduction to algebraic topology, Graduate Texts in Mathematics, vol. 119, Springer, New York, 1988.
  • J. H. C. Whitehead, Simplicial spaces, nuclei and m-groups, Proc. London Math. Soc. (2) 45 (1939), no. 1, 243–327.
  • J. H. C. Whitehead, Simple homotopy types, Amer. J. Math. 72 (1950), 1–57.
  • E. C. Zeeman, On the dunce hat, Topology 2 (1964), 341–358.