Geometry & Topology

Small values of the Lusternik–Schnirelmann category for manifolds

Alexander N Dranishnikov, Mikhail G Katz, and Yuli B Rudyak

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We prove that manifolds of Lusternik–Schnirelmann category 2 necessarily have free fundamental group. We thus settle a 1992 conjecture of Gomez-Larrañaga and Gonzalez-Acuña by generalizing their result in dimension 3 to all higher dimensions. We also obtain some general results on the relations between the fundamental group of a closed manifold M, the dimension of M and the Lusternik–Schnirelmann category of M, and we relate the latter to the systolic category of M.

Article information

Geom. Topol., Volume 12, Number 3 (2008), 1711-1727.

Received: 15 July 2007
Revised: 7 March 2008
Accepted: 5 April 2008
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55M30: Ljusternik-Schnirelman (Lyusternik-Shnirelʹman) category of a space
Secondary: 53C23: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces 57N65: Algebraic topology of manifolds

category weight cohomological dimension detecting element essential manifolds free fundamental group Lusternik–Schnirelmann category systolic category


Dranishnikov, Alexander N; Katz, Mikhail G; Rudyak, Yuli B. Small values of the Lusternik–Schnirelmann category for manifolds. Geom. Topol. 12 (2008), no. 3, 1711--1727. doi:10.2140/gt.2008.12.1711.

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