Open Access
2014 Applications of Lusternik-Schnirelmann Category and its Generalizations
John Oprea
J. Geom. Symmetry Phys. 36: 59-97 (2014). DOI: 10.7546/jgsp-36-2014-59-97


This paper explores some applications of Lusternik-Schnirelmann theory and its recent offshoots. In particular, we show how the LS category of real projective space leads to the Borsuk-Ulam theorem and the Brouwer fixed point theorem. After the development of some LS categorical tools, we also show the importance of LS category in understanding the Arnold conjecture on fixed points of Hamiltonian diffeomorphisms. We then examine ways in which LS category fits into the framework of differential geometry. In particular, we give a refinement of Bochner's theorem on the first Betti number of a non-negatively Ricci-curved space and a Bochner-like corollary to a recent theorem of Kapovitch-Petrunin-Tuschmann. Finally, we survey the new LS categorical notion of topological complexity and its relation to the motion planning problem in robotics.


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John Oprea. "Applications of Lusternik-Schnirelmann Category and its Generalizations." J. Geom. Symmetry Phys. 36 59 - 97, 2014.


Published: 2014
First available in Project Euclid: 27 May 2017

zbMATH: 1330.55002
MathSciNet: MR3379442
Digital Object Identifier: 10.7546/jgsp-36-2014-59-97

Rights: Copyright © 2014 Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences

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