Lagrangian exotic spheres
2016 (English)In: Journal of Topology and Analysis, Vol. 8, no 3, 375-397 p.Article in journal (Refereed) PublishedText
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  and  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.
Symplectic manifold; Lagrangian submanifold; floer theory; exotic sphere
IdentifiersURN: urn:nbn:se:uu:diva-289152DOI: 10.1142/S1793525316500199ISI: 000378644000001OAI: oai:DiVA.org:uu-289152DiVA: diva2:924788
FunderKnut and Alice Wallenberg FoundationSwedish Research Council, 2012-2365