Minimizing roots of maps between spheres and projective spaces in codimension one
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- by Marcio Colombo Fenille, Daciberg Lima Gonçalves and Gustavo de Lima Prado PDF
- Proc. Amer. Math. Soc. 150 (2022), 5473-5482 Request permission
Abstract:
We determine a lower bound for the dimension of the Čech cohomology of the root sets of maps from the sphere $S^{2n+1}$ and from the real projective space ${\mathbb {R}\mathrm {P}}^{2n+1}$ into the complex projective space ${\mathbb {C}\mathrm {P}}^n$, for $n\geq 1$. For each such a map, we construct a representative of its homotopy class which realize the lower bound and whose root set is minimal in the class. We prove that the circle is a minimal root set for any non-trivial homotopy class. We present analogous results for maps from both $S^{4n+3}$ and ${\mathbb {R}\mathrm {P}}^{4n+3}$ into the orbit space ${\mathbb {C}\mathrm {P}}^{2n+1}\!/\tau$, for $n\geq 0$, where $\tau$ is a free involution on ${\mathbb {C}\mathrm {P}}^{2n+1}$. In this setting, we prove that the disjoint union of two circles is a minimal root set for any non-trivial homotopy class.References
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Additional Information
- Marcio Colombo Fenille
- Affiliation: Universidade Federal de Uberlândia – Faculdade de Matemática. Av. João Naves de Ávila, 2121, Santa Mônica, 38400-902, Uberlândia MG, Brasil
- MR Author ID: 862593
- ORCID: 0000-0001-8146-3143
- Email: mcfenille@gmail.com
- Daciberg Lima Gonçalves
- Affiliation: Universidade de São Paulo – Instituto de Matemática e Estatística. Rua do Matão, 1010, Cidade Universitária, 05508-090, São Paulo SP, Brasil
- ORCID: 0000-0003-4032-7078
- Email: dlgoncal@ime.usp.br
- Gustavo de Lima Prado
- Affiliation: Universidade Federal de Uberlândia – Faculdade de Matemática. Av. João Naves de Ávila, 2121, Santa Mônica, 38400-902, Uberlândia MG, Brasil
- MR Author ID: 1258602
- ORCID: 0000-0002-6997-5573
- Email: glprado@ufu.br
- Received by editor(s): July 19, 2021
- Received by editor(s) in revised form: March 2, 2022, and March 3, 2022
- Published electronically: July 15, 2022
- Additional Notes: The second author was partially sponsored by Projeto Temático FAPESP, grant 2016/24707-4: Topologia Algébrica, Geométrica e Diferencial. The partnership of this work began during a visit of the second author to the first and third ones at Universidade Federal de Uberlândia
- Communicated by: Julie Bergner
- © Copyright 2022 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 150 (2022), 5473-5482
- MSC (2020): Primary 55M20; Secondary 55Q05
- DOI: https://doi.org/10.1090/proc/16067