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 ‘$S^1$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 $S^1$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 $S^1$S1$S^1$-equivariant contact homology with $\mathbb{Q}$$\mathbb{Q}$. We also present examples showing that the $S^1$$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.

中文翻译：

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S1$S^1$-超紧接触形式的等变接触同源性

在之前的一篇论文中，我们证明了具有有理系数的圆柱接触同调的原始定义在具有动态凸接触形式的封闭三流形上是有效的。然而，我们没有证明这种圆柱形接触同源性是接触结构的不变量。在本文中，我们定义了“非等变接触同源性”和“${\mathrm{小号}}^1$小号1$S^1$- 等变接触同调'，两者都具有整数系数，用于在任何维度上的封闭流形上的接触形式，没有可收缩的 Reeb 轨道。我们证明了这些接触同源性仅取决于接触结构。我们的构造使用 Morse-Bott 理论并且与正相关${\mathrm{小号}}^1$小号1$S^1$-Bourgeois-Oancea 的等变辛同调。然而，我们不是使用哈密顿弗洛尔同调，而是直接在接触几何中工作，使用几乎复杂的结构族。当也可以定义圆柱接触同调时，它与${\mathrm{小号}}^1$小号1$S^1$- 等变接触同源性$\mathbb{问}$$\mathbb{Q}$. 我们还提供了一些例子，表明${\mathrm{小号}}^1$$S^1$-等变接触同源性包含有趣的扭转信息。在随后的论文中，我们将使用障碍束粘合将上述故事扩展到具有动态凸接触形式的封闭三流形，这将特别证明它们的圆柱接触同源性具有仅取决于接触结构的整数系数的提升.