论文标题
莫尔斯（Morse
Morse homology for the Hamiltonian action in cotangent bundles
论文作者
论文摘要
在本文中，我们使用[10]中引入的梯度流程方程式在Cotangent Bundle $ t^*m $的封闭歧管$ M $的混合规则循环中构建了Hamiltonian Action $ \ Mathbb A_H $的摩尔斯综合体。临界点对之间的连接被实现为不稳定和稳定的歧管之间的真实交叉点，尽管它（尽管是无限的尺寸对象），但事实证明它们具有有限的尺寸交点，并具有良好的紧凑性能。这是从存在的附加结构的存在，即强烈整合的（0）式的subbundle，在哈密顿作用的负梯度流下表现得很好，并且需要进行比较。横向性是通过在一类伪级分数矢量字段中普遍扰动负梯度向量场$ - \ nabla \ nabla \ nabla \ mathbb a_h $来实现。这取决于希尔伯特歧管上对向量场的独立兴趣的抽象横向结果，其稳定和不稳定的休息点是无限的维度。由此产生的摩尔斯同源性独立于选择哈密顿（以及所有其他选择的选择），但（0）肯定的subbundle的选择，但是仅通过索引的转移而改变了莫尔斯 - 复杂性），并且与$ t^*m $的浮动同源物以及$ m $ $ m $ $ m $ $ m $的奇异同源性同源性同源。
In this paper we use the gradient flow equation introduced in [10] to construct a Morse complex for the Hamiltonian action $\mathbb A_H$ on a mixed regularity space of loops in the cotangent bundle $T^*M$ of a closed manifold $M$. Connections between pairs of critical points are realized as genuine intersections between unstable and stable manifolds, which (despite being infinite dimensional objects) turn out to have finite dimensional intersection with good compactness properties. This follows from the existence of an additional structure, namely a strongly integrable (0)-essential subbundle, which behaves nicely under the negative gradient flow of the Hamiltonian action and which is needed to make comparisons. Transversality is achieved by generically perturbing the negative gradient vector field $-\nabla \mathbb A_H$ of the Hamiltonian action within a class of pseudo-gradient vector fields preserving all good compactness properties of $-\nabla \mathbb A_H$. This follows from an abstract transversality result of independent interest for vector fields on a Hilbert manifold for which stable and unstable manifolds of rest points are infinite dimensional. The resulting Morse homology is independent of the choice of the Hamiltonian (and of all other choices but the choice of the (0)-essential subbundle, which however only changes the Morse-complex by a shift of the indices) and is isomorphic to the Floer homology of $T^*M$ as well as to the singular homology of the free loop space of $M$.
