Illinois Journal of Mathematics

Cohomology of decomposition and the multiplicity theorem with applications to dynamical systems

Leon A. Luxemburg

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The article obtains some lower bounds for the sectional category of a map based on the cohomology of the base space and the total space. We also obtain geometric results on the multiplicity of maps and show applications to equilibria on the boundary of stability regions (basins of attraction) of dynamical systems on differentiable manifolds. We consider a number of generalizations of Lusternik–Schnirelmann’s theorem which states that if a covering of an $n$ dimensional sphere consists of $n + 1$ closed sets, then at least one of the sets contains antipodal points. An elementary proof is given for a generalization of this result to packings of Euclidean spaces.

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Illinois J. Math., Volume 54, Number 2 (2010), 485-499.

First available in Project Euclid: 14 October 2011

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Zentralblatt MATH identifier

Primary: 55M30: Ljusternik-Schnirelman (Lyusternik-Shnirelʹman) category of a space
Secondary: 37D15: Morse-Smale systems 55R10: Fiber bundles


Luxemburg, Leon A. Cohomology of decomposition and the multiplicity theorem with applications to dynamical systems. Illinois J. Math. 54 (2010), no. 2, 485--499. doi:10.1215/ijm/1318598669.

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