## Geometry & Topology

### Small values of the Lusternik–Schnirelmann category for manifolds

#### 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
Revised: 7 March 2008
Accepted: 5 April 2008
First available in Project Euclid: 20 December 2017

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

#### 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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