Geometry & Topology

Small values of the Lusternik–Schnirelmann category for manifolds

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

Full-text: Open access

Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513800107

Digital Object Identifier
doi:10.2140/gt.2008.12.1711

Mathematical Reviews number (MathSciNet)
MR2421138

Zentralblatt MATH identifier
1152.55002

Subjects
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

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

Citation

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. https://projecteuclid.org/euclid.gt/1513800107


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