On the Stable Morse Number of a Closed Manifold

Damian, Mihai

doi:10.1112/s0024609301008955

Cited by 13 publications

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“…We here briefly discuss the notion of the Morse and stable Morse number. We refer to [11] for more details concerning the closed case.…”

Section: 21mentioning

confidence: 99%

“…Now let stabMorse(M ) denote the stable Morse number of a manifold M with possibly non-empty boundary, see Definition 2.5. Using Theorem 1.1 and the adaptation of [11,Theorem 2.2] to the case of manifolds with boundary (see Proposition 2.9), the following result is immediate:…”

mentioning

confidence: 99%

“…Remark 1.4. Damian's examples in [11] (see Theorem 2.6) can be used to produce a Hamiltonian for which (L × R k ) φ 1 Hs (L × R k ) is strictly less than the Morse number of L, given that k 0 is sufficiently large.…”

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confidence: 99%

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The stable Morse number as a lower bound for the number of Reeb chords

Rizell

Golovko

2018

Journal of Symplectic Geometry

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Assume that we are given a closed chord-generic Legendrian submanifold Λ ⊂ P × R of the contactisation of a Liouville manifold, where Λ moreover admits an exact Lagrangian filling L Λ ⊂ R × P × R inside the symplectisation. Under the further assumptions that this filling is spin and has vanishing Maslov class, we prove that the number of Reeb chords on Λ is bounded from below by the stable Morse number of L Λ . Given a general exact Lagrangian filling L Λ , we show that the number of Reeb chords is bounded from below by a quantity depending on the homotopy type of L Λ , following Ono-Pajitnov's implementation in Floer homology of invariants due to Sharko. This improves previously known bounds in terms of the Betti numbers of either Λ or L Λ .

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“…We here briefly discuss the notion of the Morse and stable Morse number. We refer to [11] for more details concerning the closed case.…”

Section: 21mentioning

confidence: 99%

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confidence: 99%

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The stable Morse number as a lower bound for the number of Reeb chords

Rizell

Golovko

2018

Journal of Symplectic Geometry

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“…Theorem 6.1 is proved. Estimates for Morse numbers were studied in [1,[4][5][6][7][10][11][12][13][14][15][16][17][18][19], where other approaches were used. In subsequent papers, we shall give the values of Morse numbers for manifolds of other classes.…”

Section: Applicationsmentioning

confidence: 99%

Morse functions on cobordisms

Sharko¹

2011

Ukr Math J

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“…However, the best lower bound that is known in the case when L is Lagrangian isotopic to the 0-section is in terms of the stable Morse number Morse st (M ), i.e., the minimal number of critical points of a function on M × R q which, outside a compact set, coincides with the pullback of a nondegenerate quadratic form on R q . The numbers Morse(M ) and Morse st (M ) are known to be different (see, e.g., [14]). It does not seem feasible to the author that holomorphic curve methods could be used to prove an estimate in terms of Morse(M ).…”

Section: How Far Could Arnold Conjectures Go?mentioning

confidence: 99%

Untitled

2015

Bull. Amer. Math. Soc.

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Abstract. Flexible and rigid methods coexisted in symplectic topology from its inception. While the rigid methods dominated the development of the subject during the last three decades, the balance has somewhat shifted to the flexible side in the last three years. In the talk we survey the recent advances in symplectic flexibility in the work of S. Borman, K. Cieliebak, T. Ekholm, E. Murphy, I. Smith, and the author.This survey article is an expanded version of author's article [27] and his talk at the Current Events Bulletin at the AMS meeting in Baltimore in January 2014. It also uses materials from the papers [6,9,20,32] and the book [8]. The h-principleMany problems in mathematics and its applications deal with partial differential equations, partial differential inequalities or, more generally, with partial differential relations (see [48] and also [31]), i.e., any conditions imposed on partial derivatives of an unknown function. A solution of such a partial differential relation R is any function which satisfies this relation.With any differential relation one can associate the underlying algebraic relation by substituting all the derivatives entering the relation with new independent functions. The existence of a solution of the corresponding algebraic relation, called a formal solution of the original differential relation R, is a necessary condition for the solvability of R. Though it seems that this necessary condition should be very far from being sufficient, it was a surprising discovery in the 1950s of geometrically interesting problems where existence of a formal solution is the only obstruction for the genuine solvability. Two of the first such non-trivial examples were the C 1 -isometric embedding theorem of J. Nash and N. Kuiper [57,68] and the immersion theory of S. Smale and M. Hirsch [55,77]. After Gromov's remarkable series of papers, beginning with his paper [47] and culminating in his book [48], the area crystallized as an independent subject, called the h-principle.Rigid and flexible results coexist in many areas of geometry, but nowhere else do they come so close to each other, as in symplectic topology, which serves as a rich source of examples on both sides of the flexible-rigid spectrum. Flexible and rigid problems and the development of each side toward the other shaped and continue to shape the subject of symplectic topology from its inception.

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On the Stable Morse Number of a Closed Manifold

Abstract: This paper provides examples of closed manifolds having a Morse number different from the stable Morse number. 2000 Mathematics Subject Classification 57Q10, 20F05.

Cited by 13 publications

References 15 publications

The stable Morse number as a lower bound for the number of Reeb chords

The stable Morse number as a lower bound for the number of Reeb chords

Morse functions on cobordisms

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