Geodesic
On the cohomology rings of partially projective quaternionic Stiefel manifolds
Sbornik. Mathematics, Tome 213 (2022) no. 3, pp. 300-318 Cet article a éte moissonné depuis la source Math-Net.Ru

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The quaternionic Stiefel manifold $V_{n,k}(\mathbb H)$ is the total space of a fibre bundle over the corresponding Grassmannian $G_{n,k}(\mathbb H)$. The group $\operatorname{Sp}(1)=S^3$ acts freely on the fibres of this bundle. The quotient space is called the quaternionic projective Stiefel manifold. Its real and complex analogues were actively studied earlier by a number of authors. A finite group acting freely on the three-dimensional sphere also acts freely and discretely on the fibres of the quaternionic Stiefel bundle. The corresponding quotient spaces are called partially projective Stiefel manifolds. The cohomology rings of partially projective quaternionic Stiefel manifolds with coefficients in $\mathbb Z_p$, where $p$ is prime, are calculated. Bibliography: 14 titles.
Keywords: projective Stiefel manifold, discrete free group actions on $S^3$. 
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G. E. Zhubanov; F. Yu. Popelenskii. On the cohomology rings of partially projective quaternionic Stiefel manifolds. Sbornik. Mathematics, Tome 213 (2022) no. 3, pp. 300-318. http://geodesic.mathdoc.fr/item/SM_2022_213_3_a1/

[1] A. Borel, “Sur la cohomologie des espaces fibres principaux et des espaces homogenes de groupes de Lie compacts”, Ann. of Math. (2), 57 (1953), 115–207 | DOI | MR | Zbl

[2] S. Gitler, D. Handel, “The projective Stiefel manifolds. I”, Topology, 7 (1968), 39–46 | DOI | MR | Zbl

[3] A. Borel, “Sur l'homologie et la cohomologie des groupes de Lie compacts connexes”, Amer. J. Math., 76:2 (1954), 273–342 | DOI | MR | Zbl

[4] P. F. Baum, W. Browder, “The cohomology of quotients of classical groups”, Topology, 3:4 (1965), 305–336 | DOI | MR | Zbl

[5] L. Smith, “Some remarks on projective Stiefel manifolds, immersions of projective spaces and spheres”, Proc. Amer. Math. Soc., 80:4 (1980), 663–669 | DOI | MR | Zbl

[6] L. Astey, S. Gitler, E. Micha, G. Pastor, “Cohomology of complex projective Stiefel manifolds”, Canad. J. Math., 51:5 (1999), 897–914 | DOI | MR | Zbl

[7] S. S. Gondhali, Vector fields on certain quotients of the complex Stiefel manifolds, Ph.D. thesis, Tata Inst. Fund. Res., Mumbai, 2012, 47 pp. http://math.haifa.ac.il/sgondhal/

[8] W. Threlfall, H. Seifert, “Topologische Untersuchung der Diskontinuitätsbereiche endlicher Bewegungsgruppen des dreidimensionalen sphärischen Raumes (Schluß)”, Math. Ann., 107:1 (1933), 543–586 | DOI | MR | Zbl

[9] H. Hopf, “Zum Clifford–Kleinschen Raumproblem”, Math. Ann., 95:1 (1926), 313–339 | DOI | MR | Zbl

[10] W. S. Massey, F. P. Peterson, “The cohomology structure of certain fibre spaces. I”, Topology, 4 (1965), 47–65 | DOI | MR | Zbl

[11] P. Orlik, Seifert manifolds, Lecture Notes in Math., 291, Springer-Verlag, Berlin–New York, 1972, viii+155 pp. | DOI | MR | Zbl

[12] J. Milnor, “Groups which act on $S^n$ without fixed points”, Amer. J. Math., 79:3 (1957), 623–630 | DOI | MR | Zbl

[13] S. Tomoda, P. Zvengrowski, “Remarks on the cohomology of finite fundamental groups of 3-manifolds”, The Zieschang Gedenkschrift, Geom. Topol. Monogr., 14, Geom. Topol. Publ., Coventry, 2008, 519–556 | DOI | MR | Zbl

[14] A. R. Shastri, P. Zvengrowski, “Type of 3-manifolds and addition of relativistic kinks”, Rev. Math. Phys., 3:4 (1991), 467–478 | DOI | MR | Zbl