The Symplectic Geometry of Cotangent Bundles from a Categorical Viewpoint

By K. Fukaya, P. Seidel, I. Smith. 2007

85 citations recorded by the source corpus.

We describe various approaches to understanding Fukaya categories of cotangent bundles. All of the approaches rely on introducing a suitable class of noncompact Lagrangian submanifolds. We review the work of Nadler-Zaslow [29, 30] and the authors [14], before discussing a new approach using family Floer cohomology [10] and the “wrapped Fukaya category”. The latter, inspired by Viterbo’s symplectic homology, emphasizes the connection to loop spaces, hence seems particularly suitable when trying to extend the existing theory beyond the simply connected case.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c814a234dece1fc31ed6c7949298934085c6efce

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Citations in the corpus

The monotone wrapped Fukaya category and the open-closed string map (2012, background): “The wrapped Fukaya category W(E) of a non-compact convex symplectic manifold E, introduced in [23, 8] in the exact setup, is of increasing importance both in symplectic topology (topology of Stein manifolds [29], the nearby Lagrangian conjecture [5]) and in Homological Mirror Symmetry [6, 44].”
An open string analogue of Viterbo functoriality (2007, background): “One could certainly think of applications to the chord conjecture, in line with what has been done for symplectic cohomology, and also to Lagrangian embeddings, in particular fleshing out the last part of [15].”
Effect of Legendrian surgery (2009, methodology): “We are especially th nkful (in alphabetical order) to Mohammed Abouzaid, Denis Auroux, Cheolhyun Cho, Kai Cieliebak, Jian He, Helmut Hofer, Janko Latschev, Lenny Ng, Alexandru Oance, Josh Sabloff, Eric Schoenfeld, Paul Seidel, Ivan Smith, and Alexander Voronov.”
Nearby Lagrangians with vanishing Maslov class are homotopy equivalent (2010, background): “4 (or a spectral sequence argument as in the introduction of [12]) shows that this is only possible if Q̃p is in fact quasi-isomorphic to the zero section.”
Lagrangian cobordisms via generating families: Construction and geography (2014, unknown): “Lagrangian cobordisms between Legendrian submanifolds have recently enjoyed increasing interest, partly due to their centrality in symplectic field theory (see Eliashberg, Givental and Hofer [19], and see also Bourgeois, Ekholm and Eliashberg [7] and Ekholm [13; 14] for recent examples) and wrapped ”
On the wrapped Fukaya category and based loops (2009, background): “Equivalence for cotangent fibres 30 Appendix A. Technical results 33 References 39”
Exact Lagrangian submanifolds in simply-connected cotangent bundles (2007, background): “We postpone a more detailed comparison (and a discussion of the situation when π1(N) = 0) to [6].”
Altering symplectic manifolds by homologous recombination (2010, methodology): “We will consider the Floer cohomology groups HF (Vi, Vj) taken inside the fibre M , as well as the wrapped Floer cohomology [27, 2] groups HW (∆i,∆j) in E.”
Rational SFT, Linearized Legendrian Contact Homology, and Lagrangian Floer Cohomology (2009, methodology): “Seidel [14] who explained it using an exact sequence in wrapped Floer homology [1, 13] similar to (1.”
Floer homology on the universal cover, a proof of Audin's conjecture and other constraints on Lagrangian submanifolds (2010, background): “Smith, who proved in [23], [24] that when L is relatively spin with vanishing Maslov class, its cohomology is isomorphic to the cohomology of K.”
Topological quantum field theory structure on symplectic cohomology (2010, background): “Special cases for M = T N were introduced by Abbondandolo-Schwarz [1], the general definition arises in Fukaya-Seidel-Smith [20], and a detailed construction is in Abouzaid-Seidel [3].”
Generating the Fukaya categories of Hamiltonian 𝐺-manifolds (2015, methodology): “For example, the papers of Fukaya-Seidel-Smith [26, 27] and Abouzaid [2, 4] provide concrete ways of understanding the wrapped Fukaya categories of cotangent bundles: there is a full and faithful embedding of W(T∗G; k) into the category of A∞ -bimodules over the dg-algebra C−∗(ΩG; k) of cubical chai”
Symplectic homology and the Eilenberg-Steenrod axioms (2015, background): “The Lagrangian symplectic homology group SH∗(L) coincides with the wrapped Floer homology group of L introduced in [3, 44].”