On the topological complexity of Grassmann manifolds
-
Vimala Ramani
Abstract
We prove that the topological complexity of a quaternionic flag manifold is half of its real dimension. For the real oriented Grassmann manifolds G͠n,k, 3 ≤ k ≤ [n/2], the zero-divisor cup-length of the rational cohomology of G͠n,k is computed in terms of n and k which gives a lower bound for the topological complexity of G͠n,k, TC(G͠n,k). When k = 3, it is observed in certain cases that better lower bounds for TC(G͠n,3) are obtained using ℤ2-cohomology.
Dedicated to Professor Parameswaran Sankaran on the occasion of his 60th birthday
(Communicated by Július Korbaš)
Acknowledgement
The author would like to thank Professor Parameswaran Sankaran for the encouragement and the very useful discussions. The author also would like to thank Professor Aniceto Murillo for sending a copy of his paper [15].
The author is highly indebted to the referee for the valuable comments and suggestions, in particular for pointing out a lacuna in the proof of Theorem 1.5 in the case n = 7 in the previous version; the referee’s helpful suggestions have made the proof of Theorem 1.5 thorough and clear.
References
[1] Basu, S.—Chakraborty, P.: On the cohomology ring and upper characteristic rank of Grassmannian of oriented 3-planes, J. Homotopy Relat. Struct. (2019), https://doi.org/10.1007/s40062-019-00244-1.Suche in Google Scholar
[2] Cohen, D.—Suciu, A.: Boundary manifolds of projective hypersurfaces, Adv. Math. 206 (2006), 538–566.10.1016/j.aim.2005.10.003Suche in Google Scholar
[3] Dranishnikov, A.: On topological complexity of twisted products, Topology Appl. 179 (2015), 74–80.10.1016/j.topol.2014.08.017Suche in Google Scholar
[4] Farber, M.: Topological complexity of motion planning, Discrete Comput. Geom. 29 (2003), 211–221.10.1007/s00454-002-0760-9Suche in Google Scholar
[5] Farber, M.—Tabachnikov, S.—Yuzvinsky, S.: Topological robotics: motion planning in projective spaces, https://arxiv.org/pdf/math/0210018.pdfSuche in Google Scholar
[6] Farber, M.: Instabilities of robot motion, Topology Appl. 140 (2004), 245–266.10.1016/j.topol.2003.07.011Suche in Google Scholar
[7] Farber, M.—Grant, M.—Lupton, G.—Oprea, J.: Bredon cohomology and robot motion planning, Algebr. Geom. Topol. 19 (2019), 2023–2059.10.2140/agt.2019.19.2023Suche in Google Scholar
[8] Farber, M.—Grant, M.—Lupton, G.—Oprea, J.: An upper bound for topological complexity, Topology Appl. 255 (2019), 109–125.10.1016/j.topol.2019.01.007Suche in Google Scholar
[9] Fukaya, T.: Gröbner basis for oriented Grassmann manifolds, Homology, Homotopy Appl. 10(2) (2008), 195–209.10.4310/HHA.2008.v10.n2.a10Suche in Google Scholar
[10] Korbaš, J.: Bounds for the cup-length of Poincaré spaces and their applications, Topology Appl. 153 (2006), 2976–2986.10.1016/j.topol.2006.01.005Suche in Google Scholar
[11] Korbaš, J.: The cup-length of the oriented Grassmannians vs a new bound for zero-cobordant manifolds, Bull. Belg. Math. Soc. Simon Stevin 17 (2010), 69–81.10.36045/bbms/1267798499Suche in Google Scholar
[12] Korbaš, J.: The characteristic rank and cup-length in oriented Grassmann manifolds, Osaka J. Math. 52 (2015), 1163–1172.Suche in Google Scholar
[13] Korbaš, J.—Rusín, T.: A note on the ℤ2-cohomology algebra of oriented Grassmann manifolds, Rend. Circ. Mat. Palermo (2) 65(3) (2016), 507–517.10.1007/s12215-016-0249-7Suche in Google Scholar
[14] Korbaš, J.—Rusín, T.: On the cohomology of oriented Grassmann manifolds, Homology, Homotopy Appl. 18(2) (2016), 71–84.10.4310/HHA.2016.v18.n2.a4Suche in Google Scholar
[15] Lechuga, L.—Murillo, A.: Topological complexity of formal spaces. Contemp. Math. 438, 2007, pp. 105–114.10.1090/conm/438/08448Suche in Google Scholar
[16] Mukherjee, G.—Sankaran, P.: Minimal models of oriented Grassmannians and applications, Math. Slovaca 50(5) (2000), 567–579.Suche in Google Scholar
[17] Ramani, V.—Sankaran, P.: On degrees of maps between Grassmannians, Proc. Indian Acad. Sci. Math.Sci 107(1) (1997), 13–19.10.1007/BF02840469Suche in Google Scholar
[18] Petrović, Z. Z.—Prulović, B. I.—Radovanović, M.: Characteristic rank of canonical vector bundles over oriented Grassmann manifoldsG͠3,n, Topology Appl. 230 (2017), 114–121.10.1016/j.topol.2017.08.010Suche in Google Scholar
[19] Rusin, T.: Bounds for the characteristic rank and cup-length of oriented Grassmann manifolds, Arch. Math. (Brno) 54 (2018), 313–329.10.5817/AM2018-5-313Suche in Google Scholar
[20] Tralle, A.—Oprea, J.: Symplectic Manifolds with no Kähler Structure, LNM 1661, Springer-Verlag, 1997.10.1007/BFb0092608Suche in Google Scholar
© 2020 Mathematical Institute Slovak Academy of Sciences
Artikel in diesem Heft
- Regular papers
- Monadic pseudo BE-algebras
- Quadruple construction of decomposable double MS-algebras
- Fibonacci numbers in generalized Pell sequences
- Solutions of a generalized markoff equation in Fibonacci numbers
- Root separation for polynomials with reducible derivative
- Quadratic refinements of Young type inequalities
- Sharp bounds for the Toader mean of order 3 in terms of arithmetic, quadratic and contraharmonic means
- Integration with respect to deficient topological measures on locally compact spaces
- Differential subordinations and Pythagorean means
- Analogs of Hayman’s Theorem and of logarithmic criterion for analytic vector-valued functions in the unit ball having bounded L-index in joint variables
- Unbounded oscillation of fourth order functional differential equations
- Oscillation criteria for a class of nonlinear discrete fractional order equations with damping term
- Entropy as an integral operator: Erratum and modification
- Unital topology on a unital l-group
- On the topological complexity of Grassmann manifolds
- Properties and methods of estimation for a bivariate exponentiated Fréchet distribution
- Existence and Hyers-Ulam stability results for a class of fractional order delay differential equations with non-instantaneous impulses
- Two semigroup rings associated to a finite set of meromorphic functions
Artikel in diesem Heft
- Regular papers
- Monadic pseudo BE-algebras
- Quadruple construction of decomposable double MS-algebras
- Fibonacci numbers in generalized Pell sequences
- Solutions of a generalized markoff equation in Fibonacci numbers
- Root separation for polynomials with reducible derivative
- Quadratic refinements of Young type inequalities
- Sharp bounds for the Toader mean of order 3 in terms of arithmetic, quadratic and contraharmonic means
- Integration with respect to deficient topological measures on locally compact spaces
- Differential subordinations and Pythagorean means
- Analogs of Hayman’s Theorem and of logarithmic criterion for analytic vector-valued functions in the unit ball having bounded L-index in joint variables
- Unbounded oscillation of fourth order functional differential equations
- Oscillation criteria for a class of nonlinear discrete fractional order equations with damping term
- Entropy as an integral operator: Erratum and modification
- Unital topology on a unital l-group
- On the topological complexity of Grassmann manifolds
- Properties and methods of estimation for a bivariate exponentiated Fréchet distribution
- Existence and Hyers-Ulam stability results for a class of fractional order delay differential equations with non-instantaneous impulses
- Two semigroup rings associated to a finite set of meromorphic functions
