Geodesic
Genus and the Lyusternik-Schnirelmann category of preimages
Modelirovanie i analiz informacionnyh sistem, Tome 14 (2007) no. 4, pp. 31-35.

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In this paper some axiomatic generalization (function of open subsets) of the relative Lyusternik-Schnirelmann category is considered. For this function and a given map of the underlying space to a finite polyhedron some lower bounds on the value of this function on the preimage of some point are given. 
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R. N. Karasev. Genus and the Lyusternik-Schnirelmann category of preimages. Modelirovanie i analiz informacionnyh sistem, Tome 14 (2007) no. 4, pp. 31-35. http://geodesic.mathdoc.fr/item/MAIS_2007_14_4_a5/

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