Morse Theory and the Lusternik–Schnirelmann Category of Quaternionic Grassmannians

Macías–Virgós, E.; Pereira-Sáez, M. J.; Tanré, Daniel

doi:10.1017/s0013091516000195

Cited by 3 publications

(2 citation statements)

References 14 publications

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“…In [23], the authors used Formula (2.1) directly, once a convenient function was chosen, to give the upper bound cat Sp(n) ≤ (n + 1)n/2 for the L-S category of the symplectic group. In a similar way, an optimal upper bound was given in [24] for the L-S category of the quaternionic Grassmannians…”

Section: Introductionmentioning

confidence: 99%

Homotopic Distance and Generalized Motion Planning

2022

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We prove that the homotopic distance between two maps defined on a manifold is bounded above by the sum of their subspace distances on the critical submanifolds of any Morse–Bott function. This generalizes the Lusternik–Schnirelmann theorem (for Morse functions) and a similar result by Farber for the topological complexity. Analogously, we prove that, for analytic manifolds, the homotopic distance is bounded by the sum of the subspace distances on any submanifold and its cut locus. As an application, we show how navigation functions can be used to solve a generalized motion planning problem.

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Section: Introductionmentioning

confidence: 99%

Homotopic Distance and Generalized Motion Planning

2022

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“…Different techniques of proof have been given for this result, as the use of the Cayley transform in [14], or Morse-Bott functions in [6]. Let us also mention that Morse-Bott functions are also present in [9], [13] for the study of LS-category. Finally recall the existence of a lower bound for the LS-category of Stiefel manifolds, generally better than the classical cup-length, established by Kishimoto in [7], and recalled in Theorem 4.1.…”

Section: Introductionmentioning

confidence: 99%

Relative singular value decomposition and applications to LS-category

Macías–Virgós

Pereira-Sáez

Tanré

2019

Linear Algebra and its Applications

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Let Sp(n) be the symplectic group of quaternionic (n × n)-matrices. For any 1 ≤ k ≤ n, an element A of Sp(n)In this work, starting from a singular value decomposition of P , we obtain what we call a relative singular value decomposition of A. This feature is well adapted for the study of the quaternionic Stiefel manifold X n,k , and we apply it to the determination of the Lusternik-Schnirelmann category of Sp(k) in X 2k−j,k , for j = 0, 1, 2.2010 Mathematics Subject Classification. Primary 15A33; Secondary 22E20, 22F30, 55M30.

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Morse-Bott functions on orthogonal groups

Bozma¹,

Gillam²,

Öztürk³

2019

Topology and its Applications

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We make a detailed study of various (quadratic and linear) Morse-Bott trace functions on the orthogonal groups O(n). We describe the critical loci of the quadratic trace function Tr(AXBX T ) and determine their indices via perfect fillings of tables associated with the multiplicities of the eigenvalues of A and B. We give a simplified treatment of T. Frankel's analysis of the linear trace function on SO(n), as well as a combinatorial explanation of the relationship between the mod 2 Betti numbers of SO(n) and those of the Grassmannians G(2k, n) obtained from this analysis. We review the basic notions of Morse-Bott cohomology in a simple case where the set of critical points has two connected components. We then use these results to give a new Morse-theoretic computation of the mod 2 Betti numbers of SO(n).

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Morse Theory and the Lusternik–Schnirelmann Category of Quaternionic Grassmannians

Abstract: The Lusternik–Schnirelmann category of the quaternionic Grassmannianis known to bek(n − k). In this paper we show that this result can be deduced from Morse theory.

Cited by 3 publications

References 14 publications

Homotopic Distance and Generalized Motion Planning

Homotopic Distance and Generalized Motion Planning

Relative singular value decomposition and applications to LS-category

Morse-Bott functions on orthogonal groups

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