Sectional Category and the parametrized Borsuk-Ulam property
| Autor: | Zapata, Cesar A. Ipanaque, Gonçalves, Daciberg L. |
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| Rok vydání: | 2024 |
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| Druh dokumentu: | Working Paper |
| Popis: | In this paper, for fibrations $p:E\to B$, $p':E'\to B$ over $B$ and a free involution $\tau:E\to E$ which satisfy the equality $p\circ\tau=p$, we discover a connection between the sectional category of the double covers $q:E\to E/\tau$ and $q^Y:F_{p'}(E',2)\to F_{p'}(E',2)/\mathbb{Z}_2$ from the $2$-ordered fibre-wise configuration space $F_{p'}(E',2)$ to its unordered quotient $ F_{p'}(E',2)/\mathbb{Z}_2$, and the parametrized Borsuk-Ulam property (PBUP) for the triple $\left((p,\tau);p'\right)$. Explicitly, we demonstrate that the triple $\left((p,\tau);p'\right)$ satisfies the PBUP if the sectional category of $q$ is bigger than the sectional category of $q^{p'}$. This property connects a standard problem in parametrized Borsuk-Ulam theory to current research trends in sectional category. As applictions of our results, we explore the PBUP for $E, E'$ one of the following fibrations: trivial fibration, Hopf fibration and the Fadell-Neuwirth fibration. Comment: 20 pages. Comments are welcome |
| Databáze: | arXiv |
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