Spectral numbers in Floer theories
Spectral numbers in Floer theories
复制标题
弗洛尔理论中的谱数
DOI：
10.1112/s0010437x08003564
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发表时间：
2007
影响因子：
1.8
通讯作者：
Michael Usher
中科院分区：
文献类型：
--
作者：
Michael Usher
摘要Floer同调理论的链复合物通常进行实值过滤，允许一个关联到每个Floer同调类的谱数定义为代表该类的链的过滤水平的下确界。这些谱数在Hamilton Floer同调的情况下已经被Oh，施瓦茨和其他人广泛地研究过。我们证明了与任何非零Floer同调类相关联的谱数总是有限的，并且谱数的定义中的下确界总是达到的。在哈密顿的情况下，这意味着所谓的“非退化谱性”公理在所有闭辛流形上成立。我们的证明是完全代数和适用于任何Floer型理论（包括诺维科夫同源）满足一定的标准形式的属性。关键成分是一个定理的存在性最佳逼近的任意元素的numerically生成的自由模诺维科夫环的元素规定的子模块就一定家庭的非阿基米德度量。
Abstract The chain complexes underlying Floer homology theories typically carry a real-valued filtration, allowing one to associate to each Floer homology class a spectral number defined as the infimum of the filtration levels of chains representing that class. These spectral numbers have been studied extensively in the case of Hamiltonian Floer homology by Oh, Schwarz and others. We prove that the spectral number associated to any nonzero Floer homology class is always finite, and that the infimum in the definition of the spectral number is always attained. In the Hamiltonian case, this implies that what is known as the ‘nondegenerate spectrality’ axiom holds on all closed symplectic manifolds. Our proofs are entirely algebraic and apply to any Floer-type theory (including Novikov homology) satisfying certain standard formal properties. The key ingredient is a theorem about the existence of best approximations of arbitrary elements of finitely generated free modules over Novikov rings by elements of prescribed submodules with respect to a certain family of non-Archimedean metrics.