Gröbner bases in the mod 2 cohomology of oriented Grassmann manifolds G͠2t,3
-
Uroš A. Colović
Abstract
For n a power of two, we give a complete description of the cohomology algebra H*(G͠n,3; ℤ2) of the Grassmann manifold G͠n,3 of oriented 3-planes in ℝn. We do this by finding a reduced Gröbner basis for an ideal closely related to this cohomology algebra. Using this Gröbner basis we also present an additive basis for H*(G͠n,3; ℤ2).
(Communicated by Tibor Macko)
References
[1] Bartík, V.—Korbaš, J.: Stiefel-Whitney characteristic classes and parallelizability of Grassmann manifolds, Rend. Circ. Mat. Palermo (2) 33(Suppl. 6) (1984), 19-29.Search in Google Scholar
[2] Basu, S.—Chakraborty, P.: On the cohomology ring and upper characteristic rank of Grassmannian of oriented 3-planes, J. Homotopy Relat. Struct. 15 (2020), 27-60.Search in Google Scholar
[3] Becker, T.—Weispfenning, V.: Gröbner Bases: A Computational Approach to Commutative Algebra . Grad. Texts in Math., Springer-Verlag, New York, 1993.Search in Google Scholar
[4] Borel, A.: La cohomologie mod 2 de certains espaces homogènes, Comm. Math. Helv. 27 (1953), 165–197.Search in Google Scholar
[5] Fukaya, T.: Gröbner bases of oriented Grassmann manifolds, Homol. Homotopy Appl. 10(2) (2008), 195–209.Search in Google Scholar
[6] Jaworowski, J.: An additive basis for the cohomology of real Grassmannians . Lecture Notes in Math. 1474, Springer-Verlag, Berlin, 1991, pp. 231–234.Search in Google Scholar
[7] Korbaš, J.: The characteristic rank and cup-length in oriented Grassmann manifolds, Osaka J. Math. 52 (2015), 1163–1172.Search in Google Scholar
[8] Korbaš, J.—Rusin, T.: A note on the ℤ2-cohomology algebra of oriented Grassmann manifolds, Rend. Circ. Mat. Palermo, II. Ser 65 (2016), 507–517.Search in Google Scholar
[9] Milnor, J. W.—Stasheff, J. D.: Characteristic Classes, Ann. of Math. Studies 76, Princeton University Press, New Jersey. 1974.Search in Google Scholar
© 2024 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- Prof. RNDr. Gejza Wimmer, DrSc. – 3/4 C?
- On hyper (r, q)-Fibonacci polynomials
- The pointfree version of 𝓒c(X) via the ranges of functions
- On the x-coordinates of Pell equations that are products of two Pell numbers
- Subordination properties and coefficient problems for a novel class of convex functions
- Certain radii problems for 𝓢∗(ψ) and special functions
- On direct and inverse Poletsky inequalities with a tangential dilatation
- Fourth-order nonlinear strongly non-canonical delay differential equations: new oscillation criteria via canonical transform
- Hermite interpolation of type total degree associated with certain spaces of polynomials
- Decomposition in direct sum of seminormed vector spaces and Mazur–Ulam theorem
- A fixed point technique to the stability of Hadamard 𝔇-hom-der in Banach algebras
- Compact subsets of Cλ,u(X)
- Variations of star selection principles on hyperspaces
- On lower density operators
- Gröbner bases in the mod 2 cohomology of oriented Grassmann manifolds G͠2t,3
- On the von Bahr–Esseen inequality for pairwise independent random vectors in Hilbert spaces with applications to mean convergence
- Large deviations for some dependent heavy tailed random sequences
- Metric, stratifiable and uniform spaces of G-permutation degree
- In memory of Paolo
Articles in the same Issue
- Prof. RNDr. Gejza Wimmer, DrSc. – 3/4 C?
- On hyper (r, q)-Fibonacci polynomials
- The pointfree version of 𝓒c(X) via the ranges of functions
- On the x-coordinates of Pell equations that are products of two Pell numbers
- Subordination properties and coefficient problems for a novel class of convex functions
- Certain radii problems for 𝓢∗(ψ) and special functions
- On direct and inverse Poletsky inequalities with a tangential dilatation
- Fourth-order nonlinear strongly non-canonical delay differential equations: new oscillation criteria via canonical transform
- Hermite interpolation of type total degree associated with certain spaces of polynomials
- Decomposition in direct sum of seminormed vector spaces and Mazur–Ulam theorem
- A fixed point technique to the stability of Hadamard 𝔇-hom-der in Banach algebras
- Compact subsets of Cλ,u(X)
- Variations of star selection principles on hyperspaces
- On lower density operators
- Gröbner bases in the mod 2 cohomology of oriented Grassmann manifolds G͠2t,3
- On the von Bahr–Esseen inequality for pairwise independent random vectors in Hilbert spaces with applications to mean convergence
- Large deviations for some dependent heavy tailed random sequences
- Metric, stratifiable and uniform spaces of G-permutation degree
- In memory of Paolo
