Sectional category and the fixed point property
Copyright policy )Sectional category and the fixed point property
For a Hausdorff space $X$, we exhibit an unexpected connection between the sectional number of the Fadell-Neuwirth fibration $π_{2,1}^X:F(X,2)\to X$, and the fixed point property (FPP) for self-maps on $X$. Explicitly, we demonstrate that a space $X$ has the FPP if and only if 2 is the minimal cardinality of open covers $\{U_i\}$ of $X$ such that each $U_i$ admits a continuous local section for $π_{2,1}^X$. This characterization connects a standard problem in fixed point theory to current research trends in topological robotics.
16 pages
Fixed points and coincidences in algebraic topology, Discriminantal varieties and configuration spaces in algebraic topology, FOS: Mathematics, Lyusternik-Shnirel'man category of a space, topological complexity à la Farber, topological robotics (topological aspects), Algebraic Topology (math.AT), sectional category, Mathematics - Algebraic Topology, configuration spaces, Fixed point property, Configuration spaces, Sectional category, Motion planning problem, fixed point property
Fixed points and coincidences in algebraic topology, Discriminantal varieties and configuration spaces in algebraic topology, FOS: Mathematics, Lyusternik-Shnirel'man category of a space, topological complexity à la Farber, topological robotics (topological aspects), Algebraic Topology (math.AT), sectional category, Mathematics - Algebraic Topology, configuration spaces, Fixed point property, Configuration spaces, Sectional category, Motion planning problem, fixed point property
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