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Rational homotopy of maps between certain complex Grassmann manifolds

  • Prateep Chakraborty EMAIL logo and Shreedevi K. Masuti
Published/Copyright: February 9, 2018
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 68 Issue 1

Abstract

Let Gn,k denote the complex Grassmann manifold of k-dimensional vector subspaces of ℂn. Assume l,k ≤ ⌊ n/2⌋. We show that, for sufficiently large n, any continuous map h : Gn,l → Gn,k is rationally null homotopic if (i) 1 ≤ k < l, (ii) 2 < l < k < 2(l − 1), (iii) 1 < l < k, l divides n but l does not divide k.

MSC 2010: Primary 55S37; 13A02; Secondary 57T15
Keywords: Grassmann manifold; homotopy class of maps; graded algebra homomorphism; cohomology algebra
The second author thanks Indian Statistical Institute Bangalore for providing local hospitality during the course of writing this paper. She also thanks Department of Atomic Energy, Government of India, for providing funding for her post doctoral studies during which this work is done.

Communicated by Július Korbaš

Acknowledgement

We thank Prof. P. Sankaran for suggesting the problem and many useful discussions. We thank Prof. B. Sury for giving proofs of Proposition 6.5 and Example 6.7. We also thank the referees for a careful reading of the manuscript and giving suggestions which have improved our manuscript.

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Received: 2015-11-5
Accepted: 2016-5-3
Published Online: 2018-2-9
Published in Print: 2018-2-23

© 2018 Mathematical Institute Slovak Academy of Sciences

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https://doi.org/10.1515/ms-2017-0092
Keywords for this article
Grassmann manifold; homotopy class of maps; graded algebra homomorphism; cohomology algebra

Articles in the same Issue

  2. On the number of cycles in a graph
  4. Characterization of posets for order-convergence being topological
  6. Classification of posets using zero-divisor graphs
  8. On a generalized concept of order relations on B(H)
  10. A study of stabilizers in triangle algebras
  12. Revealing two cubic non-residues in a quadratic field locally
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  16. Big mapping class groups are not acylindrically hyperbolic
  18. Applications of Henstock-Kurzweil integrals on an unbounded interval to differential and integral equations
  20. Starlike and convex functions with respect to symmetric conjugate points involving conical domain
  22. Summations of Schlömilch series containing anger function terms
  24. Some vector valued sequence spaces of Musielak-Orlicz functions and their operator ideals
  26. Nuclear operators on Cb(X, E) and the strict topology
  28. More on cyclic amenability of the Lau product of Banach algebras defined by a Banach algebra morphism
  30. Matrix generalized (θ, ϕ)-derivations on matrix Banach algebras
  32. Nonlinear ∗-Jordan triple derivations on von Neumann algebras
  34. Addendum to “A sequential implicit function theorem for the chords iteration”, Math. Slovaca 63(5) (2013), 1085–1100
  36. Comparison of density topologies on the real line
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  44. Zero-divisor graphs of lower dismantlable lattices II
  46. An identity of symmetry for the degenerate Frobenius-Euler Polynomials
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