Nearby Lagrangians with vanishing Maslov class are homotopy equivalent

By M. Abouzaid. 2010

75 citations recorded by the source corpus.

We prove that the inclusion of every closed exact Lagrangian with vanishing Maslov class in a cotangent bundle is a homotopy equivalence. We start by adapting an idea of Fukaya-Seidel-Smith to prove that such a Lagrangian is equivalent to the zero section in the Fukaya category with integral coefficients. We then study an extension of the Fukaya category in which Lagrangians equipped with local systems of arbitrary dimension are admitted as objects, and prove that this extension is generated, in the appropriate sense, by local systems over a cotangent fibre. Whenever the cotangent bundle is simply connected, this generation statement is used to prove that every closed exact Lagrangian of vanishing Maslov index is simply connected. Finally, we borrow ideas from coarse geometry to develop a Fukaya category associated to the universal cover, allowing us to prove the result in the general case.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/63565c9080550d32b891f8fce487c04e36be10e8

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Citations in the corpus

The monotone wrapped Fukaya category and the open-closed string map (2012, background): “This outline is precisely the argument of Abouzaid [2], so the only difference for our setup is that we need to construct CO, ∆ and prove the equality of C1,C2 in the monotone setting.”
Parametrized Ring-Spectra and the Nearby Lagrangian Conjecture (2011, background): “Later in [4], Abouzaid removed the assumption on N and strengthened this to a homotopy equivalence.”
An algebraic approach to virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves (2013, methodology): “M/ to the coefficient ring (as in Fukaya [28] or Abouzaid [2]) to obtain a sharper lower bound in (10.”
Quantization of conic Lagrangian submanifolds of cotangent bundles (2012, result): “4) ko γ ⋆ F −→ kγ ⋆ F −→ F +1 −→, kγ ⋆ F −→ T∗(F ) −→ kγ [1] ⋆ F +1 −→ .”
A Beginner’s Introduction to Fukaya Categories (2013, unknown): “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 ∗S4k+1 [8].”
Generating the Fukaya categories of Hamiltonian 𝐺-manifolds (2015, background): “Over this field, the Koszul twisted complex is ∆[3] 0 99K ∆”