AI 中文总结
为余辛流形发展等变上同调理论，研究对称在拓扑中的作用及与泊松几何联系。重新解释Guillemin等人的障碍理论，引入等变障碍类，证明首个障碍类消失与相关泊松结构等变幺模性等价，为特定流形构建等变上同调并建立形式性结果。
AI 中文摘要
余辛流形为余维数为一的辛叶状结构提供自然几何框架，在几何和数学物理中广泛出现。我们为余辛流形发展了一种等变上同调理论，研究对称性在其拓扑中的作用以及它们与泊松几何的联系。重新解释Guillemin、Miranda和Pires的障碍理论，我们引入了用于保持叶状结构的群作用的等变障碍类，并刻画了不变余辛结构的存在性。我们进一步表明，第一个障碍类的消失等同于相关泊松结构的等变幺模性，通过Ginzburg的等变泊松上同调框架将经典准则扩展到等变情形。对于在\(\mathbb{S}^1\)上纤维化的紧致余辛流形，我们构造了德拉姆、叶状和泊松上同调的等变版本，通过等变Wang序列在每种情况下建立形式性结果。关键输入是Kirwan关于辛纤维上哈密顿作用的形式性定理，它通过等变Wang序列，根据纤维上的单值作用给出了所有三种等变上同调理论的完整且可计算的描述。
英文摘要
Cosymplectic manifolds provide a natural geometric framework for codimension-one symplectic foliations and arise throughout geometry and mathematical physics. We develop an equivariant cohomological theory for cosymplectic manifolds, studying the role of symmetry in their topology and their connections to Poisson geometry. Reinterpreting the obstruction theory of Guillemin, Miranda, and Pires, we introduce equivariant obstruction classes for group actions preserving the foliation and characterize the existence of invariant cosymplectic structures. We further show that the vanishing of the first obstruction class is equivalent to equivariant unimodularity of the associated Poisson structure, extending a classical criterion to the equivariant setting via Ginzburg's framework for equivariant Poisson cohomology. For compact cosymplectic manifolds fibering over $\mathbb{S}^1$, we construct equivariant versions of de Rham, foliated, and Poisson cohomologies, establishing formality results in each case via an equivariant Wang sequence. The key input is Kirwan's formality theorem for Hamiltonian actions on the symplectic fiber, which, through the equivariant Wang sequence, yields a complete and computable description of all three equivariant cohomology theories in terms of the monodromy action on the fiber.
Comments29 pages. Minor additions to sections 1, 2, and 4.2
