参数化环谱及其附近的拉格朗日猜想
Parametrized Ring-Spectra and the Nearby Lagrangian Conjecture
摘要：
本文证明了连通余切丛T*N中的任意闭连通正合拉格朗日流形L上到一个有限覆盖空间提升一个同调等价。我们证明了这一点，通过构造一个由流形N参数化的环谱FL的非线性参数化族。FL的同调是T*L的（扭）辛上同调。双调性意味着存在一个收敛于FL的同调的Serre谱序列，并且与N上的交积组合的积在这个谱序列上诱导出一个积。这个乘积结构和它与L上的交积的关系然后被用来获得结果。结合Abouzaid的工作，我们得到了L -> N总是同伦等价的结论.
英文摘要：
We prove that any closed connected exact Lagrangian manifold L in a connected cotangent bundle T*N is up to a finite covering space lift a homology equivalence. We prove this by constructing a fibrant parametrized family of ring spectra FL parametrized by the manifold N. The homology of FL will be (twisted) symplectic cohomology of T*L. The fibrancy property will imply that there is a Serre spectral sequence converging to the homology of FL and the product combined with intersection product on N induces a product on this spectral sequence. This product structure and its relation to the intersection product on L is then used to obtain the result. Combining this result with work of Abouzaid we arrive at the conclusion that L -> N is always a homotopy equivalence.
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