On the loop homology of complex projective spaces

Chataur, David; Borgne, Jean-François Le

doi:10.24033/bsmf.2616

Cited by 5 publications

(3 citation statements)

References 9 publications

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“…By Menichi [27], this class does not contain the even-dimensional spheres S 2m , m ≥ 1 for coefficients of characteristic zero, for instance, and the same is true for CP n , n ≥ 1, HP n , n ≥ 1 and OP 2 by results of Yang [43], Chataur-Le Borgne [11], Hepworth [17], and Cadek-Moravec [9]. Moreover, by [9,11,17] the same is true for CP n , HP n , where n is even, and for OP 2 , with F 2 -coefficients. By a result of Vaintrob [39], this class does not contain the closed surface Σ g of genus g, for each g > 1, and any choice of coefficients.…”

Section: Introductionmentioning

confidence: 86%

Symplectic cohomology and a conjecture of Viterbo

Shelukhin¹

2019

Preprint

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We identify a class of closed smooth manifolds for which there exists a uniform bound on the Lagrangian spectral norm of Hamiltonian deformations of the zero section in a unit cotangent disk bundle. This class of manifolds is characterized in topological terms involving the Chas-Sullivan algebra and the BV-operator on the homology of the free loop space. In particular, it contains spheres and is closed under products. This settles a conjecture of Viterbo from 2007 as the special case of T n . We discuss generalizations and applications.

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Section: Introductionmentioning

confidence: 86%

Symplectic cohomology and a conjecture of Viterbo

Shelukhin¹

2019

Preprint

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“…A number of calculations have been made for specific classes of manifolds (see e.g. [CJY04,Men09,Tam06,Vai07,Hep10,CLB11]). In this paper we consider manifolds that are highly connected relative to their dimension.…”

Section: Introductionmentioning

confidence: 99%

Free loop space homology of highly connected manifolds

Berglund

Börjeson

2016

Forum Mathematicum

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Abstract. We calculate the homology of the free loop space of (n − 1)-connected closed manifolds of dimension at most 3n − 2 (n ≥ 2), with the Chas-Sullivan loop product and loop bracket. Over a field of characteristic zero, we obtain an expression for the BV-operator. We also give explicit formulas for the Betti numbers, showing they grow exponentially. Our main tool is the connection between formality, coformality and Koszul algebras that was elucidated by the first author [Ber14a].

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“…Over a field F p , if p = 0, this BV algebra H * (LM ) is hard to compute. It has been computed only for complex Stiefel manifolds [40], spheres [33], compact Lie groups [19,34] and complex projective spaces [5,17].…”

Section: Introductionmentioning

confidence: 99%

The bv algebra in string topology of classifying spaces

Kuribayashi,

Menichi

2016

Preprint

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For almost any compact connected Lie group G and any field Fp, we compute the Batalin-Vilkovisky algebra H * +dim G (LBG; Fp) on the loop cohomology of the classifying space introduced by Chataur and the second author. In particular, if p is odd or p = 0, this Batalin-Vilkovisky algebra is isomorphic to the Hochschild cohomology HH * (H * (G), H * (G)). Over F 2 , such isomorphism of Batalin-Vilkovisky algebras does not hold when G = SO(3) or G = G 2 .

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On the loop homology of complex projective spaces

Cited by 5 publications

References 9 publications

Symplectic cohomology and a conjecture of Viterbo

Symplectic cohomology and a conjecture of Viterbo

Free loop space homology of highly connected manifolds

The bv algebra in string topology of classifying spaces

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