Small values of the Lusternik-Schnirelmann category for manifolds

TL;DR

This paper proves that manifolds with Lusternik-Schnirelmann category 2 have free fundamental groups, confirming a longstanding conjecture and exploring relationships between fundamental groups, dimension, and categories of manifolds.

Contribution

It generalizes a 1992 conjecture to all higher dimensions, establishing a key link between LS category and fundamental group properties.

Findings

Manifolds with LS category 2 have free fundamental groups

Confirmed the 1992 conjecture of Gomez-Larranaga and Gonzalez-Acuna

Connected LS category with systolic category of manifolds

Abstract

We prove that manifolds of Lusternik-Schnirelmann category 2 necessarily have free fundamental group. We thus settle a 1992 conjecture of Gomez-Larranaga and Gonzalez-Acuna, 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 relate the latter to the systolic category of M.