Symplectic cohomology and a conjecture of Viterbo

We identify a new 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 settles a well-known conjecture of Viterbo from 2007 as the special case of T n , which has...

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Bibliographic Details
Published inGeometric and functional analysis Vol. 32; no. 6; pp. 1514 - 1543
Main Author Shelukhin, Egor
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.12.2022
Springer Nature B.V
Subjects
Analysis
Homology
Manifolds (mathematics)
Mathematics
Mathematics and Statistics
Operators (mathematics)
Topology
Loop space
53D40
Viterbo conjecture
Spectral norm
37J05
53D12
Symplectic cohomology
Lagrangian submanifold
symplectic topology
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Summary:We identify a new 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 settles a well-known conjecture of Viterbo from 2007 as the special case of T n , which has been completely open for n > 1 . Our methods are different and more intrinsic than those of the previous work of the author first settling the case n = 1 . The new class of manifolds is defined in topological terms involving the Chas–Sullivan algebra and the BV-operator on the homology of the free loop space. It contains spheres and is closed under products. We discuss generalizations and various applications, to C 0 symplectic topology in particular.
Bibliography:ObjectType-Article-1
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ISSN:1016-443X
1420-8970
DOI:10.1007/s00039-022-00619-2