Free $\mathbb {Z}^p$-actions on the three dimensional torus
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- by Eduardo Fierro Morales and Richard Urzúa-Luz PDF
- Proc. Amer. Math. Soc. 149 (2021), 2871-2878 Request permission
Abstract:
We show that for each natural $p\geq 2$, the Lefschetz fixed point theorem is optimal when applied to $\mathbb {Z}^{p}$-actions by homeomorphisms on the three dimensional torus $\mathbb {T}^3$. More precisely, we show that for a spectrally unitary $\mathbb {Z}^p$-action ${\mathbf {A}}$ on the first homology group $H_1(\mathbb {T}^3,\mathbb {Z} )$ with trivial fixed point set, there exists a free $\mathbb {Z}^p$-action by real analytic diffeomorphisms of $\mathbb {T}^3$ whose induced $\mathbb {Z}^p$-action on $H_1(\mathbb {T}^3,\mathbb {Z})$ is the action ${\mathbf {A}}$. In particular, we establish the normal form for this type of actions.References
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Additional Information
- Eduardo Fierro Morales
- Affiliation: Universidad Católica del Norte, Casilla 1280, Antofagasta, Chile
- ORCID: 0000-0002-6931-4046
- Email: efierro@ucn.cl
- Richard Urzúa-Luz
- Affiliation: Universidad Católica del Norte, Casilla 1280, Antofagasta, Chile
- ORCID: 0000-0003-2230-565X
- Email: rurzua@ucn.cl
- Received by editor(s): May 22, 2017
- Received by editor(s) in revised form: June 11, 2019, and November 11, 2019
- Published electronically: April 7, 2021
- Additional Notes: The first author was partially supported by DGIP of the Universidad Católica del Norte, Antofagasta, Chile.
The second author was partially supported by Fondecyt # 1100832 and DGIP of the Universidad Católica del Norte, Antofagasta, Chile. - Communicated by: Nimish Shah
- © Copyright 2021 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 149 (2021), 2871-2878
- MSC (2020): Primary 37B05, 57M60, 57R30
- DOI: https://doi.org/10.1090/proc/15191
- MathSciNet review: 4257801