## Illinois Journal of Mathematics

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

Leon A. Luxemburg

#### Abstract

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.

#### Article information

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

Dates
First available in Project Euclid: 14 October 2011

https://projecteuclid.org/euclid.ijm/1318598669

Digital Object Identifier
doi:10.1215/ijm/1318598669

Mathematical Reviews number (MathSciNet)
MR2846470

Zentralblatt MATH identifier
1246.55002

#### Citation

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. https://projecteuclid.org/euclid.ijm/1318598669

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