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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 \(n+1\).

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
Full Text: DOI
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