Geodesic
On Actions of Tori and Quaternionic Tori on Products of Spheres
Informatics and Automation, Topology, Geometry, Combinatorics, and Mathematical Physics, Tome 326 (2024), pp. 5-14 Cet article a éte moissonné depuis la source Math-Net.Ru

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We study the actions of tori (standard compact tori as well as their quaternionic analogs) on products of spheres. We prove that the orbit space of a specific action of a torus on a product of spheres is homeomorphic to a sphere. A similar statement for the real torus $\mathbb Z_2^n$ was proved by the second author in 2019. We also extend this result to arbitrary compact topological groups, thus generalizing the results mentioned above as well as the results of the first author on the actions of a compact torus of complexity $1$.
Mots-clés : torus action, quaternions, orbit space. 
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     author = {Anton A. Ayzenberg and Dmitry V. Gugnin},
     title = {On {Actions} of {Tori} and {Quaternionic} {Tori} on {Products} of {Spheres}},
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Anton A. Ayzenberg; Dmitry V. Gugnin. On Actions of Tori and Quaternionic Tori on Products of Spheres. Informatics and Automation, Topology, Geometry, Combinatorics, and Mathematical Physics, Tome 326 (2024), pp. 5-14. http://geodesic.mathdoc.fr/item/TRSPY_2024_326_a0/

[1] V. I. Arnold, “Relatives of the quotient of the complex projective plane by the complex conjugation”, Proc. Steklov Inst. Math., 224 (1999), 46–56 | MR | Zbl

[2] Atiyah M., Berndt J., “Projective planes, Severi varieties and spheres”, Surveys in differential geometry, Harvard Univ., 2002, v. VIII, Surv. Differ. Geom., 8, Lectures on geometry and topology held in honor of Calabi, Lawson, Siu, and Uhlenbeck, International Press, Somerville, MA, 2003, 1–27 | DOI | MR | Zbl

[3] A. A. Ayzenberg, “Torus actions of complexity 1 and their local properties”, Proc. Steklov Inst. Math., 302 (2018), 16–32 | DOI | DOI | MR | Zbl

[4] Ayzenberg A., “Torus action on quaternionic projective plane and related spaces”, Arnold Math. J., 7:2 (2021), 243–266 | DOI | MR | Zbl

[5] Ayzenberg A., Masuda M., “Orbit spaces of equivariantly formal torus actions of complexity one”, Transform. Groups, 2023 | DOI | Zbl

[6] G. E. Bredon, Introduction to Compact Transformation Groups, Pure Appl. Math., 46, Academic Press, New York, 1972 | MR | Zbl

[7] Buchstaber V.M., Panov T.E., Toric topology, Math. Surv. Monogr., 204, Am. Math. Soc., Providence, RI, 2015 | DOI | MR | Zbl

[8] Gorchakov V., Equivariantly formal 2-torus actions of complexity one, E-print, 2023, arXiv: 2304.00936 [math.AT] | DOI

[9] D. V. Gugnin, “Branched coverings of manifolds and $nH$-spaces”, Funct. Anal. Appl., 53:2 (2019), 133–136 | DOI | DOI | MR | Zbl

[10] D. V. Gugnin, “On nonfree actions of commuting involutions on manifolds”, Math. Notes, 113:5–6 (2023), 770–775 | DOI | DOI | MR | Zbl

[11] M. A. Mikhaĭlova, “On the quotient space modulo the action of a finite group generated by pseudoreflections”, Math. USSR, Izv., 24:1 (1985), 99–119 | DOI | MR | Zbl | Zbl

[12] Neofytidis C., Non-zero degree maps between manifolds and groups presentable by products, PhD thesis, Ludwig Maximilian Univ., Munich, 2014 https://edoc.ub.uni-muenchen.de/17204/1/Neofytidis\textunderscore Christoforos.pdf

[13] Neofytidis C., “Branched coverings of simply connected manifolds”, Topology Appl., 178 (2014), 360–371 | DOI | MR | Zbl

[14] O. G. Styrt, “On the orbit space of a compact linear Lie group with commutative connected component”, Trans. Moscow Math. Soc., 2009 (2009), 171–206 | DOI | MR | Zbl