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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 $$n+1$$.

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
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