Czechoslovak Mathematical Journal

Czechoslovak Mathematical Journal, Vol. 70, No. 2, pp. 299-310, 2020

On real flag manifolds with cup-length equal to its dimension

Marko Radovanović

Received June 14, 2018. Published online November 18, 2019.

Abstract: We prove that for any positive integers $n_1,n_2,\ldots,n_k$ there exists a real flag manifold $F(1,\ldots,1,n_1,n_2,\ldots,n_k)$ with cup-length equal to its dimension. Additionally, we give a necessary condition that an arbitrary real flag manifold needs to satisfy in order to have cup-length equal to its dimension.

Keywords: cup-length; flag manifold; Lyusternik-Shnirel'man category

Classification MSC: 14M15, 55M30, 57N65

DOI: 10.21136/CMJ.2019.0283-18

PDF available at: Springer Institute of Mathematics CAS Digital Mathematics Library

References:
[1] I. Berstein: On the Lusternik-Schnirelmann category of Grassmannians. Math. Proc. Camb. Philos. Soc. 79 (1976), 129-134. DOI 10.1017/S0305004100052142 | MR 0400212 | Zbl 0315.55011
[2] A. Borel: La cohomologie mod 2 de certains espaces homogènes. Comment. Math. Helv. 27 (1953), 165-197. (In French.) DOI 10.1007/BF02564561 | MR 0057541 | Zbl 0052.40301
[3] H. L. Hiller: On the cohomology of real Grassmanians. Trans. Am. Math. Soc. 257 (1980), 521-533. DOI 10.2307/1998310 | MR 0552272 | Zbl 0462.57021
[4] L'. Horanská, J. Korbaš: On cup products in some manifolds. Bull. Belg. Math. Soc. - Simon Stevin 7 (2000), 21-28. DOI 10.36045/bbms/1103055716 | MR 1741743 | Zbl 0956.55010
[5] J. Korbaš, J. Lörinc: The $\mathbb Z_2$-cohomology cup-length of real flag manifolds. Fundam. Math. 178 (2003), 143-158. DOI 10.4064/fm178-2-4 | MR 2029922 | Zbl 1052.55006
[6] Z. Z. Petrović, B. I. Prvulović, M. Radovanović: On maximality of the cup-length of flag manifolds. Acta Math. Hung. 149 (2016), 448-461. DOI 10.1007/s10474-016-0625-y | MR 3518647 | Zbl 1389.57006
[7] Z. Z. Petrović, B. I. Prvulović, M. Radovanović: Gröbner bases for (partial) flag manifolds. To appear in J. Symb. Comput. DOI 10.1016/j.jsc.2019.06.008
[8] M. Radovanović: Gröbner bases for some flag manifolds and applications. Math. Slovaca 66 (2016), 1065-1082. DOI 10.1515/ms-2016-0204 | MR 3602604 | Zbl 1399.13029
[9] M. Radovanović: On the $\Bbb Z_2$-cohomology cup-length of some real flag manifolds. Filomat 30 (2016), 1577-1590. DOI 10.2298/FIL1606577R | MR 3530103 | Zbl 06749816
[10] R. E. Stong: Cup products in Grassmannians. Topology Appl. 13 (1982), 103-113. DOI 10.1016/0166-8641(82)90012-8 | MR 0637432 | Zbl 0469.55005

Affiliations: Marko Radovanović, University of Belgrade, Faculty of Mathematics, Studentski trg 16, Belgrade, Serbia, e-mail: markor@matf.bg.ac.rs

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