The Cohomology of Projective Unitary Groups

Haibao Duan

Informatics and Automation, Topology, Geometry, Combinatorics, and Mathematical Physics, Tome 326 (2024), pp. 173-192

Résumé

The projective unitary group $\mathrm {PU}(n)$ is the quotient of the unitary group $\mathrm {U}(n)$ by its center $S^1=\{e^{i\theta }I_n: \theta \in [0,2\pi ]\}$, where $I_n$ is the identity matrix. Combining the Serre spectral sequence of the fibration $\mathrm {PU}(n)\to \mathrm {PU}(n)/T$ with the Gysin sequence of the circle bundle $\mathrm {U}(n)\to \mathrm {PU}(n)$, we compute the integral cohomology ring of $\mathrm {PU}(n)$ using explicitly constructed generators, where $T$ is a maximal torus of $\mathrm {PU}(n)$.

Détail

Keywords: Lie groups, cohomology, Serre spectral sequence
Mots-clés : Gysin sequence.

Bibliographie

[1] Antieau B., Williams B., “The topological period–index problem over 6-complexes”, J. Topol., 7:3 (2014), 617–640 | DOI | MR | Zbl

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

[3] Atiyah M., “$K$-theory past and present”, Proceedings of the Berlin Mathematical Society. Years 1997–2000, Berl. Math. Ges., Berlin, 2001, 411–417 ; arXiv: math/0012213 [math.KT] | MR | Zbl | DOI

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

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

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

[7] Bott R., Samelson H., “The cohomology ring of $G/T$”, Proc. Natl. Acad. Sci. USA, 41:7 (1955), 490–493 | DOI | MR | Zbl

[8] Can M.B., The Betti numbers of a determinantal variety, E-print, 2019, arXiv: 1903.06849 [math.AG] | DOI | MR

[9] Duan H., “On the Borel transgression in the fibration $G\to G/T$”, Homology Homotopy Appl., 20:1 (2018), 79–86 | DOI | MR | Zbl

[10] Duan H., Schubert calculus in Lie groups, E-print, 2023, arXiv: 2307.13904 [math.AT] | DOI

[11] Duan H., Lin X., “Topology of unitary groups and the prime orders of binomial coefficients”, Sci. China. Math., 60:9 (2017), 1543–1548 | DOI | MR | Zbl

[12] Duan H., Zhao X., “Schubert calculus and the Hopf algebra structures of exceptional Lie groups”, Forum. math., 26:1 (2014), 113–139 | DOI | MR | Zbl

[13] Duan H., Zhao X., “Schubert presentation of the cohomology ring of flag manifolds $G/T$”, LMS J. Comput. Math., 18:1 (2015), 489–506 | DOI | MR | Zbl

[14] H. Duan and X. Zhao, “Schubert calculus and intersection theory of flag manifolds”, Russ. Math. Surv., 77:4 (2022), 729–751 | DOI | DOI | MR | Zbl

[15] Grothendieck A., “Torsion homologique et sections rationnelles”, Anneaux de Chow et applications: Sem. C. Chevalley, v. 3, Exp. 5, Secr. math., Paris, 1958, 1–29

[16] Grove L.C., Classical groups and geometric algebra, Grad. Stud. Math., 39, Am. Math. Soc., Providence, RI, 2002 | MR

[17] Hirsch G., “L'anneau de cohomologie d'un espace fibré en sphères”, C. r. Acad. sci. Paris, 241 (1955), 1021–1023 | MR | Zbl

[18] Hopf H., “Über die Topologie der Gruppen-Mannigfaltigkeiten und ihre Verallgemeinerungen”, Ann. Math. Ser. 2, 42:1 (1941), 22–52 | DOI | MR | Zbl

[19] Kac V.G., “Torsion in cohomology of compact Lie groups and Chow rings of reductive algebraic groups”, Invent. math., 80:1 (1985), 69–79 | DOI | MR | Zbl

[20] Leray J., “Sur l'homologie des groupes de Lie, des espaces homogènes et des espaces fibrés principaux”, Colloque de topologie (espaces fibrés), Bruxelles, 1950, Centre Belge Rech. Math., G. Thone, Liège, 1951, 101–115 | MR

[21] Massey W.S., “On the cohomology ring of a sphere bundle”, J. Math. Mech., 7 (1958), 265–290 | MR

[22] Mathai V., Rosenberg J., “Group dualities, T-dualities, and twisted $K$-theory”, J. London Math. Soc. Ser. 2, 97:1 (2018), 1–23 | DOI | MR | Zbl

[23] McCleary J., A user's guide to spectral sequences, Cambridge Stud. Adv. Math., 58, 2nd ed., Cambridge Univ. Press, Cambridge, 2001 | DOI | MR

[24] Milnor J.W., Stasheff J.D., Characteristic classes, Ann. Math. Stud., 76, Princeton Univ. Press, Princeton, NJ, 1974 | DOI | MR | Zbl

[25] Petrie T., “The $K$ theory of the projective unitary groups”, Bull. Am. Math. Soc., 72 (1966), 875–878 | DOI | MR | Zbl

[26] Petrie T., “The $K$ theory of the projective unitary groups”, Topology, 6 (1967), 103–115 | DOI | MR | Zbl

[27] Pittie H.V., “The integral homology and cohomology rings of $\mathrm {SO}(n)$ and $\mathrm {Spin}(n)$”, J. Pure Appl. Algebra, 73:2 (1991), 105–153 | DOI | MR | Zbl

[28] Ruiz C.A., “The cohomology of the complex projective Stiefel manifold”, Trans. Am. Math. Soc., 146 (1969), 541–547 | DOI | MR | Zbl

[29] Weil A., Foundations of algebraic geometry, AMS Colloq. Publ., 29, Am. Math. Soc., Providence, RI, 1962 | MR | Zbl

[30] Whitehead G.W., Elements of homotopy theory, Grad. Texts Math., 61, Springer, Berlin, 1978 | DOI | MR | Zbl