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Lusternik-Schnirelmann Category and Related Topics

The Lusternik-Schnirelmann category of a topological spaces is a numerical invariant defined in a very simple way. There are several equivalent definitions.

  • definition by open covering
  • definition by fat wedge
  • Ganea’s definition by fibre-cofibre sequence
  • inductive definition by using mapping cones

The original motivation was to count the number of critical points of a smooth function on a manifold . The Lusternik-Schnierelmann category is still used in geometric problems. For example, Rudyak and Schlenk [ RS ] studied the minimal number of charts on a symplectic manifold .

There are several variations. For example,

  • strong category

Farber [ Far ] introduced a variant cat( X ; ξ ) depending on real 1-dimenional cohomology classes ξ H 1 ( X ; ). Farber studies this invariants in [ Far04 FSa FSb ] .

Colman [ Col ] proposes a definition for Lie groupoids .

The definition can be extended to objects in model categories . For example, [ Doe93 HL94 Kah03 GCGD ] .

There are many related numerical invariants.


[Col]     Hellen Colman. The Lusternik-Schnirelmann category of a Lie groupoid, arXiv:0908.3325 .

[DKR]     Alexander N. Dranishnikov, Mikhail G. Katz, and Yuli B. Rudyak. Systolic category, cohomological dimension, and self-linking, arXiv:0807.5040 .

[Doe93]     Jean-Paul Doeraene. L.S.-category in a model category. J. Pure Appl. Algebra , 84(3):215–261, 1993.

[DR]     Alexander N. Dranishnikov and Yuli B. Rudyak. Stable Systolic Category of Manifolds and the Cup-length, arXiv:0812.4637 .

[Far]     Michael Farber. Zeros of closed 1-forms, homoclinic orbits, and Lusternik - Schnirelman theory, arXiv:math.DG/0106046 .

[Far04]     Michael Farber. Topology of closed one-forms , volume 108 of Mathematical Surveys and Monographs . American Mathematical Society, Providence, RI, 2004.

[FSa]     Michael Farber and Dirk Schütz. Cohomological estimates for cat( X,ξ ), arXiv:math.AT/0609005 .

[FSb]     Michael Farber and Dirk Schütz. Homological category weights and estimates for cat 1 ( X,ξ ), arXiv:math.AT/0609141 .

[GCGD]     J. M. Garcia-Calcines and P. R. Garcia-Diaz. Inductive Lusternik-Schnirelmann category in a model category, arXiv:math/0612619 .

[HL94]     Kathryn P. Hess and Jean-Michel Lemaire. Generalizing a definition of Lusternik and Schnirelmann to model categories. J. Pure Appl. Algebra , 91(1-3):165–182, 1994.

[Kah03]     Thomas Kahl. On the algebraic approximation of Lusternik-Schnirelmann category. J. Pure Appl. Algebra , 181(2-3):227–277, 2003.

[KR]     Mikhail G. Katz and Yuli B. Rudyak. Lusternik-Schnirelmann category and systolic category of low dimensional manifolds, arXiv:math/0410456 .

[KV]     R.N. Karasev and A.Yu. Volovikov. Configuration-like spaces and coincidences of maps on orbits, arXiv:0911.4338 .

[RS]     Yuli Rudyak and Felix Schlenk. Minimal atlases of closed symplectic manifolds, arXiv:math.SG/0605350 .