Reidemeister classes for coincidences between sections of a fiber bundle
Let $s_0,f_0$ be two sections of a fiber bundle $q: E\to B$ and the coincidence set $\Gamma(s_0,f_0)\neq \emptyset$. We consider the following question: Is there $s_0\simeq_B s_1$ (by the homotopies which cover the constant homotopy $\overline{I}_B$ on the basic space) such that $\Gamma(s_1, f_0)=\...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Sociedade Brasileira de Matemática
2019-05-01
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| Series: | Boletim da Sociedade Paranaense de Matemática |
| Subjects: | |
| Online Access: | https://periodicos.uem.br/ojs/index.php/BSocParanMat/article/view/37223 |
| _version_ | 1797633493245624320 |
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| author | Dirceu Penteado Thales Fernando Vilamaior Paiva |
| author_facet | Dirceu Penteado Thales Fernando Vilamaior Paiva |
| author_sort | Dirceu Penteado |
| collection | DOAJ |
| description | Let $s_0,f_0$ be two sections of a fiber bundle $q: E\to B$ and the coincidence set $\Gamma(s_0,f_0)\neq \emptyset$. We consider the following question: Is there $s_0\simeq_B s_1$ (by the homotopies which cover the constant homotopy $\overline{I}_B$ on the basic space) such that $\Gamma(s_1, f_0)=\emptyset$? If $b_0\in \Gamma(s_0,f_0)$ and $F_0=q^{-1}(b_0)$ is the typical fiber, in this context we can use the homotopy lifting extension propriety of the fibration $q$ to obtain homotopies over $B$. When we make this and the basic point are fixed we can use the elements $s_0(\beta), f_0(\beta^{-1})$ where $\beta \in\pi_1(B,b_0)$ and the elements $\gamma\in \pi_1(F_0,e_0)$. So we will introduce the algebraic classes of Reidemeister relative to the subgroup $\pi_1(F_0,e_0)$. When the basic points are not fixed we need to consider the classes $[\til{s}]_L$ of lifting of $s_0$ defined on the universal covering $\til{B}$ to $\til{E}$. The present work relates the lifting classes $[\til{s}]_L$ of $s_0$ and the algebraic relative Reidemeister classes $R_A(s_0,f_0; \pi_1(F_0,e_0).$ |
| first_indexed | 2024-03-11T11:54:47Z |
| format | Article |
| id | doaj.art-65334928599c4518870158662e4fcf22 |
| institution | Directory Open Access Journal |
| issn | 0037-8712 2175-1188 |
| language | English |
| last_indexed | 2024-03-11T11:54:47Z |
| publishDate | 2019-05-01 |
| publisher | Sociedade Brasileira de Matemática |
| record_format | Article |
| series | Boletim da Sociedade Paranaense de Matemática |
| spelling | doaj.art-65334928599c4518870158662e4fcf222023-11-08T20:07:15ZengSociedade Brasileira de MatemáticaBoletim da Sociedade Paranaense de Matemática0037-87122175-11882019-05-0138610.5269/bspm.v38i6.37223Reidemeister classes for coincidences between sections of a fiber bundleDirceu Penteado0Thales Fernando Vilamaior Paiva1Universidade Federal de São Carlos Departamento de MatemáticaUniversidade Federal de Mato Grosso do Sul CPAqLet $s_0,f_0$ be two sections of a fiber bundle $q: E\to B$ and the coincidence set $\Gamma(s_0,f_0)\neq \emptyset$. We consider the following question: Is there $s_0\simeq_B s_1$ (by the homotopies which cover the constant homotopy $\overline{I}_B$ on the basic space) such that $\Gamma(s_1, f_0)=\emptyset$? If $b_0\in \Gamma(s_0,f_0)$ and $F_0=q^{-1}(b_0)$ is the typical fiber, in this context we can use the homotopy lifting extension propriety of the fibration $q$ to obtain homotopies over $B$. When we make this and the basic point are fixed we can use the elements $s_0(\beta), f_0(\beta^{-1})$ where $\beta \in\pi_1(B,b_0)$ and the elements $\gamma\in \pi_1(F_0,e_0)$. So we will introduce the algebraic classes of Reidemeister relative to the subgroup $\pi_1(F_0,e_0)$. When the basic points are not fixed we need to consider the classes $[\til{s}]_L$ of lifting of $s_0$ defined on the universal covering $\til{B}$ to $\til{E}$. The present work relates the lifting classes $[\til{s}]_L$ of $s_0$ and the algebraic relative Reidemeister classes $R_A(s_0,f_0; \pi_1(F_0,e_0).$https://periodicos.uem.br/ojs/index.php/BSocParanMat/article/view/37223Coincidence theoryReidemeister classesfiber bundle |
| spellingShingle | Dirceu Penteado Thales Fernando Vilamaior Paiva Reidemeister classes for coincidences between sections of a fiber bundle Boletim da Sociedade Paranaense de Matemática Coincidence theory Reidemeister classes fiber bundle |
| title | Reidemeister classes for coincidences between sections of a fiber bundle |
| title_full | Reidemeister classes for coincidences between sections of a fiber bundle |
| title_fullStr | Reidemeister classes for coincidences between sections of a fiber bundle |
| title_full_unstemmed | Reidemeister classes for coincidences between sections of a fiber bundle |
| title_short | Reidemeister classes for coincidences between sections of a fiber bundle |
| title_sort | reidemeister classes for coincidences between sections of a fiber bundle |
| topic | Coincidence theory Reidemeister classes fiber bundle |
| url | https://periodicos.uem.br/ojs/index.php/BSocParanMat/article/view/37223 |
| work_keys_str_mv | AT dirceupenteado reidemeisterclassesforcoincidencesbetweensectionsofafiberbundle AT thalesfernandovilamaiorpaiva reidemeisterclassesforcoincidencesbetweensectionsofafiberbundle |