The étale sectional number is either 1 or infinity
Cesar A. Ipanaque Zapata
TL;DR
The paper investigates the étale sectional number in spaces endowed with the étale quasi Grothendieck topology, establishing a sharp dichotomy: the invariant is always either 1 or ∞. For any map $f:X→Y$, if $Étale-sec(f)$ is finite, then it must be 1, and $f$ is locally sectionable; if $f$ is not locally sectionable, then $Étale-sec(f)=∞$. It further shows that for path-connected spaces $X$, the étale topological complexity $TC_{étale}(X)$ is 1 precisely when $X$ is locally contractible, and ∞ otherwise. The results link local geometric properties to global étale-section invariants, and the section discusses the interplay between étale covers, examples like the shrinking wedge of circles, and open questions about null-homotopic étale covers of spheres.
Abstract
In this work, we show that the étale sectional number (\text{Étale-sec$(-)$}), i.e., the sectional number in the category of topological spaces with the étale quasi Grothendieck topology (as defined in arXiv:2410.22515), is either 1 or infinity. Specifically, given a continuous map $f:X\to Y$, we demonstrate that \[\text{Étale-sec$(f)$}=\begin{cases} 1,&\hbox{ if $f$ is locally sectionable,} \infty,&\hbox{ if $f$ is not locally sectionable.} \end{cases} \] Additionally, for a path-connected space $X$, the étale topological complexity satisfies \[\text{TC}_{\text{étale}}(X)=\begin{cases} 1,&\hbox{ if $X$ is locally contractible,} \infty,&\hbox{ if $X$ is not locally contractible.} \end{cases} \] These results provide a way to understand the \aspas{complexity} of maps and spaces within the context of the étale quasi Grothendieck topology, a structure that considers local behavior of maps and spaces. The classification into values of 1 or infinity reflects a dichotomy in the local geometric structure of the map or space, with the presence or absence of local sections or contractibility significantly influencing the outcome.