The étale sectional number is either 1 or infinity
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arXiv
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| Format: | Preprint |
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2025
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| author | Zapata, Cesar A. Ipanaque |
| author_facet | Zapata, Cesar A. Ipanaque |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_11942 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The étale sectional number is either 1 or infinity Zapata, Cesar A. Ipanaque Algebraic Topology 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. |
| title | The étale sectional number is either 1 or infinity |
| topic | Algebraic Topology |
| url | https://arxiv.org/abs/2503.11942 |