Bogatyj, S. A. Lyusternik-Shnirel’man theorem and \({\beta} f\). (Russian. English summary) Zbl 0967.55004 Fundam. Prikl. Mat. 4, No. 1, 11-38 (1998). In the paper of J. M. Aarts, R. J. Fokkink, and H. Vermeer [Topology 35, No. 4, 1051–1056 (1996; Zbl 0918.54011)] unimprovable estimates for the power of a free covering in the terms of the dimension of a space were given. The author shows that these results can be extended to the case of a finite family of free maps as well as to more general classes of spaces. In particular, it is proved that for every \(k\) free homeomorphisms of an \(n\)-dimensional paracompact space onto itself, the coloring number does not exceed \(n+2k+1\). The author gives also generalization of H. Steinlein’s result [Can. Math. Bull. 27, 192–204 (1984; Zbl 0507.55003)] proving that for any free map of a compact space into itself, the coloring number is not greater than the quadruplicate Hopf number. Reviewer’s remark: In his remark while proofreading the author notes that some results of the paper were obtained independently by M. A. van Hartskamp and J. Vermeer [Topology Appl. 73, No. 2, 181–190 (1996; Zbl 0867.55003)]. Reviewer: Valerii V. Obukhovskii (Voronezh) Cited in 1 Review MSC: 55M30 Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects) 55M35 Finite groups of transformations in algebraic topology (including Smith theory) 55M10 Dimension theory in algebraic topology Keywords:Lyusternik-Schnirelman theorem; free covering; \(\beta f\) mapping; genus of covering; coloring number Citations:Zbl 0507.55003; Zbl 0531.55002; Zbl 0918.54011; Zbl 0867.55003 PDFBibTeX XMLCite \textit{S. A. Bogatyj}, Fundam. Prikl. Mat. 4, No. 1, 11--38 (1998; Zbl 0967.55004) Full Text: Link