Examples of cohomology manifolds which are not homologically locally connected

Symmetry groups of non-simply-connected four-manifolds

Let $M$ be a closed, connected, orientable topological four-manifold with $H_1(M)$ nontrivial and free abelian, $b_2(M)\ne 0, 2$, and $χ(M)\ne 0$. We show that if $G$ is a finite group of 2-rank $\le 1$ which admits a homologically trivial, locally linear, effective action on $M$, then $G$ must be cyclic. With additional assumptions to ensure orientability of some components of the singular set (e.g. if $G$ acts by symplectic symmetries, or preserving a spin structure), we also rule out $C_2 \times C_2$ actions. The proofs use equivariant cohomology, localization, and a careful study of the fi…

4-流形上的有限群作用与等变丛

Finite group actions on 4-manifolds and equivariant bundles

本研究针对具有同调平凡局部线性循环群作用的4-流形，通过扭曲符号公式推导得出等变丛存在的充要条件，该条件体现为不动点数据与迷向表示间的同余关系。

Homotopy and cohomology of spaces of locally convex curves in the sphere

We discuss the homotopy type and the cohomology of spaces of locally convex parametrized curves gamma: [0,1] -> S^2, i.e., curves with positive geodesic curvature. The space of all such curves with gamma(0) = gamma(1) = e_1 and gamma'(0) = gamma'(1) = e_2 is known to have three connected components X_{-1,c}, X_1, X_{-1}. We show several results concerning the homotopy type and cohomology of these spaces. In particular, X_{-1,c} is contractible, X_1 and X_{-1} are simply connected, pi_2(X_{-1}) contains a copy of Z and pi_2(X_1) contains a copy of Z^2. Also, H^n(X_{1}, R) and H^n(X_{-1}, R) are…

On piecewise smooth cohomology of locally trivial Lie groupoids

Mishchenko-Oliveira proved the piecewise smooth cohomology and Lie algebroid cohomology of a Lie algebroid on a combinatorial compact manifold are isomorphic. In this paper, we describe an application of that result locally trivial Lie groupoids on combinatorial manifolds to obtain an isomorphism between de Rham cohomology and piecewise de Rham cohomology of left invariants forms.

Morita cohomology and homotopy locally constant sheaves

We identify Morita cohomology, which is a categorification of the cohmology of a topological space X, with the category of homotopy locally constant sheaves of perfect complexes on X.

局部紧致集上层上同调的计算技术

Computational techniques for sheaf cohomology of locally profinite sets

本文提出k-sheer划分用于计算局部紧致集上常系数Z_2的上同调，通过非平凡顶上同调元作为coboundaries的极限，将中间上同调问题转化为顶上同调问题。

Comparison of sheaf cohomology and singular cohomology

A classic result states that on any locally contractible and paracompact topological space, singular cohomology and sheaf cohomology are isomorphic. A result by Ramanan claims that the paracompactness assumption may be removed, but Ramanan's proof in fact implicitly relies on the assumption of paracompactness and does not work in the stated generality. This paper gives a modified proof that, in fact, singular cohomology and sheaf cohomology agree on any locally contractible topological space.

Cohomology theories on locally conformal symplectic manifolds

In this note we introduce primitive cohomology groups of locally conformal symplectic manifolds $(M^{2n}, ω, θ)$. We study the relation between the primitive cohomology groups and the Lichnerowicz-Novikov cohomology groups of $(M^{2n}, ω, θ)$, using and extending the technique of spectral sequences developed by Di Pietro and Vinogradov for symplectic manifolds. We discuss related results by many peoples, e.g. Bouche, Lychagin, Rumin, Tseng-Yau, in light of our spectral sequences. We calculate the primitive cohomology groups of a $(2n+2)$-dimensional locally conformal symplectic nilmanifold as …

Morse-Novikov cohomology of locally conformally Kähler surfaces

We review the properties of the Morse-Novikov cohomology and compute it for all known compact complex surfaces with locally conformally Kähler metrics. We present explicit computations for the Inoue surfaces $\mathcal{S}^0$, $\mathcal{S}^+$, $\mathcal{S}^-$ and classify the locally conformally Kähler (and the tamed locally conformally symplectic) forms on $\mathcal{S}^0$. We prove the nonexistence of LCK metrics with potential and more generally, of $d_θ$-exact LCK metrics on Inoue surfaces and Oeljeklaus-Toma manifolds.

Four-Manifolds which admit Z_p x Z_p actions

We show that the simply-connected four-manifolds which admit locally linear, homologically trivial actions by rank two finite abelian groups are homeomorphic to connected sums of CP^2, -CP^2, and S^2 x S^2 (with one exception: pseudofree Z_3 x Z_3 actions on the Chern manifold), and also establish an equivariant decompostion theorem. This generalizes results from a 1970 paper by Orlik and Raymond, and complements more recent work of Fintushel, Yoshida, and Huck on S^1 actions. In each case, the simply-connected four-manifolds which support such actions are essentially the same.

局部几何偏序集的Orlik–Solomon代数与复阿贝尔排列的上同调

The Orlik--Solomon algebra of a locally geometric poset and cohomology of complex abelian arrangements

该研究构造了局部几何偏序集的Orlik–Solomon代数，利用其与复阿贝尔李群子群排列的关联，结合拓扑信息计算排列补集的有理上同调，尤其给出紧李群情形下的显式微分分次代数。

局部自由等距李群作用的谱序列

A Spectral Sequence for locally free isometric Lie group actions

该研究提出适用于自由等距李代数及局部自由等距李群作用的谱序列，关联流形德拉姆上同调、李代数上同调与基本上同调，还引入新的德拉姆上同调描述，并将Wang长正合序列推广到低余维李代数作用。

关于局部紧致集的上同调

On cohomology of locally profinite sets

本文在Zermelo-Fraenkel集合论中构造了首个非递降且忠实平坦的交换环之间映射的例子，涉及局部紧致集的上同调结构。

具有局部标准作用的开环面流形的等变上同调环

Equivariant cohomology ring of open torus manifolds with locally standard actions

该研究针对具有局部标准作用的开环面流形，采用新方法证明了其等变上同调环同构于轨道空间拓扑面环的公式无需自由部分为平凡环面丛的限制，并给出了等变Stiefel-Whitney类与Pontrjagin类的显式公式。

The quotient criterion for syzygies in equivariant cohomology for elementary abelian $2$-group actions

Let $G$ be a elementary abelian $2$-group and $X$ be a manifold with a locally standard action of $G$. We provide a criterion to determine the syzygy order of the $G$-equivariant cohomology of $X$ with coefficients over a field of characteristic two using a complex associated to the cohomology of the face filtration of the manifold with corners $X/G$. This result is the real version of the quotient criterion for locally standard torus actions developed by M. Franz in arXiv:1205.4462.

局部以$T^{n}\times S^{1}$在$T^{*}\mathbb{C}^{n}$上的作用为模型的GKM图及其图等变上同调

GKM graph locally modeled by $T^{n}\times S^{1}$-action on $T^{*}\mathbb{C}^{n}$ and its graph equivariant cohomology

该研究引入一类包含两种4n维带$T^{n}\times S^{1}$作用流形GKM图的带枝标号图，计算其图等变上同调环，借助剥壳结构给出模基并研究乘法结构。

A Cocycle Model for Topological and Lie Group Cohomology

We propose a unified framework in which the different constructions of cohomology groups for topological and Lie groups can all be treated on equal footings. In particular, we show that the cohomology of "locally continuous" cochains (respectively "locally smooth" in the case of Lie groups) fits into this framework, which provides an easily accessible cocycle model for topological and Lie group cohomology. We illustrate the use of this unified framework and the relation between the different models in various applications. This includes the construction of cohomology classes characterizing the…

Cohomology Rings of a Class of Torus Manifolds

Torus manifolds are topological generalization of smooth projective toric manifolds. We compute the rational cohomology ring of a class of smooth locally standard torus manifolds whose orbit space is a connected sum of simple polytopes.

Cohomology of the infinite-order jet space

De Rham cohomology, $d_V$- and $d_H$-cohomology of the differential algebra of locally pull-back exterior forms on the infinite-order jet manifold of a smooth fibre bundle are calculated.

局部对称品种的热带化

Tropicalizations of locally symmetric varieties

本文研究局部对称品种的热带化，探讨其在模空间和算术群上同调中的应用。

Bredon 层上同调

Bredon sheaf cohomology

本文提出Bredon层上同调这一新的等变上同调理论，用以描述有限群作用下局部紧Hausdorff空间等变层的代数K-理论与连续函数C*-代数的等变E-理论，建立其性质并证明强唯一性定理，推广了Efimov与Clausen的相关结果。

Brown representability for exterior cohomology and cohomology with compact supports

It is well known that cohomology with compact supports is not a homotopy invariant but only a proper homotopy one. However, as the proper category lacks of general categorical properties, a Brown representability theorem type does not seem reachable. However, by proving such a theorem for the so called exterior cohomology in the complete and cocomplete exterior category, we show that the $n$-th cohomology with compact supports of a given countable, locally finite, finite dimensional relative CW-complex $(X,\mathbb{R}_+)$ is naturally identified with the set $[X,K_n]^{\mathbb{R}_+}$ of exterior…

局部对称簇的余秩谱序列

Corank spectral sequence for locally symmetric varieties

本文为局部对称簇上同调的混合 Hodge 结构构造了一种新型谱序列，该序列收敛于 Hodge 三角形的边缘分量并在特定范围内于 E1 处退化，从而简化了部分 Hodge 分量的表达并导出全纯欧拉数的恒等式。

Morse-Novikov cohomology of closed one-forms of rank 1

We discuss the Morse-Novikov cohomology of a compact manifold, associated to a closed one--form whose free abelian group generated by its periods $\langle \int_γη\mid [γ] \in π_1(M)\rangle$ is of rank 1, the focus being on locally conformally symplectic manifolds. In particular, we provide an explicit computation for the Inoue surface $\mathcal{S}^0$.