Sectional category and The Fixed Point Property

TL;DR

This paper reveals a surprising link between the sectional number of a specific fibration in topology and the fixed point property of spaces, providing a new characterization of FPP via open covers and local sections.

Contribution

It establishes an exact equivalence between the fixed point property and the minimal sectional number of a Fadell-Neuwirth fibration, connecting fixed point theory with topological robotics.

Findings

Spaces with FPP have sectional number 2 for the fibration

FPP characterized by minimal open covers admitting local sections

Connects fixed point theory to topological robotics research

Abstract

For a Hausdorff space $X$, we exhibit an unexpected connection between the sectional number of the Fadell-Neuwirth fibration $\pi_{2,1}^X:F(X,2)\to X$, 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 $\pi_{2,1}^X$. This characterization connects a standard problem in fixed point theory to current research trends in topological robotics.