Abstract
For a Hausdorff space $X$, we exhibit an unexpected connection between the sectional number of the Fadell-Neuwirth fibration $\pi_{2,1}^X\colon F(X,2)\to X$, 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 $\pi_{2,1}^X$. This characterization connects a standard problem in fixed point theory to current research trends in topological robotics.
Citation
Cesar A. Ipanaque Zapata. Jesús González. "Sectional category and the fixed point property." Topol. Methods Nonlinear Anal. 56 (2) 559 - 578, 2020. https://doi.org/10.12775/TMNA.2020.033
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