#### Volume 12, issue 3 (2008)

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Small values of the Lusternik–Schnirelmann category for manifolds

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

Geometry & Topology 12 (2008) 1711–1727
##### 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 $M$.

##### Keywords
category weight, cohomological dimension, detecting element, essential manifolds, free fundamental group, Lusternik–Schnirelmann category, systolic category
##### Mathematical Subject Classification 2000
Primary: 55M30
Secondary: 53C23, 57N65