×

zbMATH — the first resource for mathematics

On colorings of maps. (English) Zbl 0867.55003
Summary: A fixed-point free map \(f:X\to X\) is said to be colorable with \(k\) colors if there exists a closed cover \({\mathcal C}\) of \(X\) consisting of \(k\) elements such that \(C\cap f(C)= \emptyset\) for every \(C\) in \({\mathcal C}\). It is shown that every fixed-point free continuous selfmap of a compact space \(X\) with \(\dim X\leq n\) can be colored with \(n+3\) colors. Similar results are obtained for finitely many maps. It is shown that every free \(\mathbb{Z}_p\)-action on an \(n\)-dimensional compact space \(X\) has genus at most \(n+1\).

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aarts, J.M.; Nishiura, T., Dimension and extensions, (1992), North-Holland Amsterdam · Zbl 0873.54037
[2] J.M. Aarts, R.J. Fokkink and J. Vermeer, Variations on a theorem of Lusternik and Schnirelmann, Topology, to appear.
[3] Błaszczyk, A.; Kim Dok Yong, A topological version of a combinatorial theorem of katĕtov, Comm. math. univ. carolinae, 29, 657-663, (1988) · Zbl 0687.54011
[4] Błaszczyk, A.; Vermeer, J., Some old and some new results on combinatorial properties of fixed-point free maps, (), to appear · Zbl 0919.54034
[5] de Bruijn, N.G.; Erdös, P., A colour problem for infinite graphs and a theorem in the theory of relations, (), 369-373 · Zbl 0044.38203
[6] van Douwen, E.K., βX and fixed-point free maps, Topology appl., 51, 191-195, (1993) · Zbl 0792.54037
[7] Krasnosel’ski, M.A., On computation of the rotation of a vectorfield on a finite dimensional spheres, Dokl. akad. nauk SSSR, 101, 401-404, (1955), (in Russian)
[8] Munkholm, H.J., Borsuk-Ulam theorems for proper \(Zp\)-action on (mod p homology) n->spheres, Math. scand., 24, 167-185, (1969) · Zbl 0186.57501
[9] Steinlein, H., Borsuk-Ulam Sätze und abbildungen mit kompakten iterierten, Dissertationes math., 177, 116, (1980) · Zbl 0439.47045
[10] Wallman, H., Lattices and topological spaces, Ann. of math., 39, 112-126, (1938) · JFM 64.0603.01
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.