Framed bordism and Lagrangian embeddings of exotic spheres

By M. Abouzaid. 2008

56 citations recorded by the source corpus.

In dimensions congruent to 1 modulo 4, we prove that the cotangent bundle of an exotic sphere which does not bound a parallelisable manifold is not symplectomorphic to the cotangent bundle of the standard sphere. More precisely, we prove that such an exotic sphere cannot embed as a Lagrangian in the cotangent bundle of the standard sphere. The main ingredients of the construction are (1) the fact that the graph of the Hopf bration embeds the standard sphere, and hence any Lagrangian which embeds in its cotangent bundle, as a displaceable Lagrangian in the product a symplectic vector space of the appropriate dimension with its complex projective space, and (2) a moduli space of solutions to a perturbed CauchyRiemann equation introduced by Gromov.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7a5dd928066e015b3dec2badb7f84dab67781da4

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Citations in the corpus

A Beginner’s Introduction to Fukaya Categories (2013, background): “Abouzaid has further shown that Floer theory detects more than purely topological information about exact Lagrangians in cotangent bundles: certain exotic spheres (in dimensions ≥ 9) do not admit Lagrangian embeddings into T S [8].”
Gauged Floer Theory Of Toric Moment Fibers (2010, methodology): “Relationship to Floer cohomology in the quotient 65 References 67”
An algebraic approach to virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves (2013, background): “r than a branched manifold). Also, as presented here, our VFC package seems inadequate for applications of moduli spaces of J-holomorphic curves involving their smooth structure (e.g. as in Abouzaid [Abo12]), though the notion of an implicit atlas (with smooth structure) and our deformations to the ”
A topological introduction to knot contact homology (2012, background): “nt and conormal bundles. Only the disk bundle portion DT∗Mof T∗M is shown, with boundary ST∗M. Note that both LK and the zero section Mare Lagrangian in T∗M, and their intersection is K. Theorem 1.2 ([Abo12]). If Σ is an exotic (4k+ 1)-sphere that does not bound a parallelizable manifold, then T∗Σ i”
Knot contact homology (2011, background): “1 is its application to an interesting general question in symplectic topology: to what extent do symplectic- and contact-geometric objects naturally associated to objects in smooth topology remember the underlying smooth topology? More specifically, how much of the smooth topology is encoded by hol”
Contact homology and virtual fundamental cycles (2015, methodology): “wledge, standard in the ﬁeld (perhaps the most similar published result is Hutchings–Taubes [HT07, HT09]). We learned these methods (while writing the gluing theorems in [Par16, §§B–C]) from Abouzaid [Abo12], McDuﬀ–Salamon [MS04, MS94], and discussions with Mohammed Abouzaid, Tobias Ekholm, Helmut H”