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Lagrangian exotic spheres
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.ORCID iD: 0000-0003-2618-5712
Univ Cambridge, Ctr Math Sci, Wilberforce Rd, Cambridge CB3 0WB, England.
2016 (English)In: Journal of Topology and Analysis, Vol. 8, no 3, 375-397 p.Article in journal (Refereed) PublishedText
Abstract [en]

Let k > 2. We prove that the cotangent bundles T*Sigma and T*Sigma' of oriented homotopy (2k -1)-spheres Sigma and Sigma' are symplectomorphic only if [Sigma] = [+/-Sigma'] is an element of Theta(2k-1)/bP(2k), where Theta(2k-1) denotes the group of oriented homotopy (2k -1)-spheres under connected sum, bP(2k) denotes the subgroup of those that bound a parallelizable 2k-manifold, and where -Sigma denotes Sigma with orientation reversed. We further show that if n = 4k -1 and RPn#Sigma admits a Lagrangian embedding in CPn, then [Sigma#Sigma] is an element of bP(4k). The proofs build on [1] and [18] in combination with a new cut-and-paste argument; that also yields some interesting explicit exact Lagrangian embeddings, for instance of the sphere S-n into the plumbing T*Sigma(n)#T-pl*Sigma(n) of cotangent bundles of certain exotic spheres. As another application, we show that there are re-parametrizations of the zero-section in the cotangent bundle of a sphere that are not Hamiltonian isotopic (as maps rather than as submanifolds) to the original zero-section.

Place, publisher, year, edition, pages
2016. Vol. 8, no 3, 375-397 p.
Keyword [en]
Symplectic manifold; Lagrangian submanifold; floer theory; exotic sphere
National Category
Geometry
Identifiers
URN: urn:nbn:se:uu:diva-289152DOI: 10.1142/S1793525316500199ISI: 000378644000001OAI: oai:DiVA.org:uu-289152DiVA: diva2:924788
Funder
Knut and Alice Wallenberg FoundationSwedish Research Council, 2012-2365
Available from: 2016-04-29 Created: 2016-04-29 Last updated: 2016-08-02Bibliographically approved

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Ekholm, TobiasKragh, Thomas
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