S1$S^1$‐equivariant contact homology for hypertight contact forms
S1$S^1$‐equivariant contact homology for hypertight contact forms
复制标题
S1$S^1$−超紧接触形式的等变接触同源性
DOI：
10.1112/topo.12240
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发表时间：
2022
影响因子：
1.1
通讯作者：
Nelson, Jo
中科院分区：
文献类型：
--
作者：
Hutchings, Michael;Nelson, Jo
在前文中，我们证明了原有理数的柱面接触同调的定义在具有动态凸接触形式的闭三流形上是有效的。然而，我们没有证明这种柱面接触同调是接触结构的不变量。本文定义了任意维闭流形上没有可压缩Reeb轨道的接触形式的“非等变接触同调”和“S_1$S^1$-等变接触同调”，这两个定义都是整系数的。我们证明了这些接触同调仅依赖于接触结构。我们的构造使用Morse-Bott理论，并与Bourgeis-Oancea的正的S_1$S^1$等变辛同调有关。然而，我们不是研究哈密顿Floer同调，而是直接研究接触几何，使用几乎复杂的结构族。当柱面接触同调也可以定义时，它与S_1$S^1$-等变接触同调Q${\mathbb{q}}$的张量积一致。我们还给出的例子表明，S1$S^1$等变接触同调包含有趣的挠率信息。在接下来的文章中，我们将利用障碍丛粘合将上述故事推广到具有动态凸接触形式的闭三流形，这将特别证明它们的柱面接触同调具有仅取决于接触结构的整数系数。
In a previous paper, we showed that the original definition of cylindrical contact homology, with rational coefficients, is valid on a closed three‐manifold with a dynamically convex contact form. However, we did not show that this cylindrical contact homology is an invariant of the contact structure. In the present paper, we define ‘nonequivariant contact homology’ and ‘S1$S^1$‐equivariant contact homology’, both with integer coefficients, for a contact form on a closed manifold in any dimension with no contractible Reeb orbits. We prove that these contact homologies depend only on the contact structure. Our construction uses Morse–Bott theory and is related to the positive S1$S^1$‐equivariant symplectic homology of Bourgeois‐Oancea. However, instead of working with Hamiltonian Floer homology, we work directly in contact geometry, using families of almost complex structures. When cylindrical contact homology can also be defined, it agrees with the tensor product of the S1$S^1$‐equivariant contact homology with Q${\mathbb {Q}}$. We also present examples showing that the S1$S^1$‐equivariant contact homology contains interesting torsion information. In a subsequent paper, we will use obstruction bundle gluing to extend the above story to closed three‐manifolds with dynamically convex contact forms, which in particular will prove that their cylindrical contact homology has a lift to integer coefficients which depends only on the contact structure.