Strong cohomological rigidity of toric varieties
Suyoung Choi, Seonjeong Park
每个具有第二贝蒂数$2$的两个非奇异完全环面簇和拟流形之间的上同调环同构分别可由一个环同态和同胚实现。
Every cohomology ring isomorphism between two non-singular complete toric varieties and quasitoric manifolds, respectively, with second Betti number $2$ is realizable by a diffeomorphism and homeomorphism, respectively.
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We show the strong cohomological rigidity of Hirzebruch surface bundles over Bott manifolds. As a corollary, we have that the strong cohomological rigidity conjecture is true for Bott manifolds of dimension $8$.
We show that the strong cohomological rigidity conjecture for Bott manifolds is true. Namely, any graded cohomology ring isomorphism between two Bott manifolds is induced by a diffeomorphism.
A Counterexample to Cohomological Rigidity of Toric Manifolds
构造两个复维数四的环面流形，它们非同伦等价但整系数上同调环同构，从而给出上同调刚性的反例。
It is proved that if two quasitoric manifolds of dimension $\le 2p^2-4$ for a prime $p$ have isomorphic cohomology rings, then they have the same $p$-local stable homotopy type.
We formulate the notion of an isomorphism of GKM graphs. We then show that two GKM graphs have isomorphic graph equivariant cohomology algebras if and only if the graphs are isomorphic.
A quasitoric manifold is a smooth manifold with a locally standard torus action for which the orbit space is identified with a simple polytope. For a class of topological spaces, the class is called strongly cohomologically rigid if any isomorphism of cohomology rings can be realized as a homeomorphism. This paper shows the strong cohomological rigidity of the class of quasitoric manifolds over $I^3$.
Strong cohomological rigidity of a product of projective spaces
证明了对于复曲面流形$M$，任何分次环同构$H^\ast(M) \to H^\ast(\prod_{i=1}^{m}\CP^{n_i})$都是由一个分次环同构$\prod_{i=1}^m \CP^{n_i} \to M$诱导的.
By Hantzsche-Wendt manifold (for short HW-manifold) we understand any oriented closed Riemannian manifold of dimension n with a holonomy group (Z_2)^{n-1}. Two HW-manifolds M_1 and M_2 are cohomological rigid if and only if a homeomorphism between M_1 and M_2 is equivalent to an isomorphism of graded rings H^{*}(M_1,F_2) and H^{*}(M_2,F_2). We prove that HW-manifolds are cohomological rigid.
Cohomological rigidity of smooth toric Fano fourfolds
该研究证明整上同调环分次同构的光滑环面法诺四维簇必微分同胚，确立其微分同胚类由上同调环决定的刚性性质。
Properties of Bott manifolds and cohomological rigidity
复曲面流形的上同调刚性问题是关于复曲面流形的上同调环是否决定了流形的拓扑型。本文考虑了单扭Bott流形类的问题，得到了该问题的肯定答案。我们还将结果推广到拟流形。在这样做时，我们表明，扭曲数的Bott流形是定义良好的，等于上同调复杂性的上同调环的流形。我们还证明了任何上同调的Bott流形同胚的Bott流形。所有这些结果也推广到具有$\mathbb Z_{(2)}$ -系数的情形，其中$\mathbb Z_{(2)}$是在2处的局部化环。
Toric cohomological rigidity of simple convex polytopes
一个简单凸多胞体$P$是\emph{cohomologically rigid}，如果它的组合结构由$P$上的拟流形的上同调环决定。不是每个$P$都有这个性质，但是一些重要的多面体如单形或立方体是上同调刚性的。本文研究了多胞形的上同调刚性，并对几类新的多胞形，包括单形的乘积，建立了上同调刚性。$P$的上同调刚性与它的\emph{Stanley--Reisner ring}的\emph{bigraded Betti numbers}有关，后者是组合交换代数中的另一个重要不变量.
On the homotopy types of $4$-dimensional toric orbifolds
本文研究4维托里克轨道的同调刚性问题，引入proper isomorphism概念，证明每个proper isomorphism类最多包含两种同伦类型，并在特殊情况下两种分类一致。
Degrees of Maps and Cohomological Rigidity of Partial Flag Manifolds
该研究证明不同复或四元数部分旗流形间若上域非Grassmann流形则连续映射Brouwer次数为零，推出这类流形具上同调刚性，并猜想其具同调刚性。
The cohomological rigidity problem for toric manifolds asks whether toric manifolds are diffeomorphic (or homeomorphic) if their integral cohomology rings are isomorphic. Many affirmative partial solutions to the problem have been obtained and no counterexample is known. In this paper, we study the diffeomorphism classification of toric Fano $d$-folds with $d=3,4$ or with Picard number $\ge 2d-2$. In particular, we show that those manifolds except for two toric Fano $4$-folds are diffeomorphic if their integral cohomology rings are isomorphic. The exceptional two toric Fano $4$-folds (their ID…
Cohomological non-rigidity of generalized real Bott manifolds of height 2
本文研究了两个高度为2的广义实Bott流形具有同构的Z/2系数上同调环和同构的情形。结果表明，具有Z/2系数的上同调环一般不区分那些直到同构的流形。这给出了\cite{ka-ma08}中实环面流形的上同调刚性问题的一个反例。我们还证明了高度为2的广义实Bott流形是同伦等价的。
(Filtered) cohomological rigidity of Bott towers
Bott塔是一个点上的迭代$\CP ^1$ -丛，其中每个$\CP ^1$ -丛是秩$2$可分解复向量丛的投影化。对于Bott塔，自然定义了过滤上同调。我们证明了Bott塔的同构类可以通过它们的滤上同调环来区分。我们甚至表明，任何过滤上同调环同构之间的两个Bott塔是由同构的Bott塔。
We introduce the category of {\it locally $k$-standard $T$-manifolds} which includes well-known classes of manifolds such as toric and quasitoric manifolds, good contact toric manifolds and moment-angle manifolds. They are smooth manifolds with well-behaved actions of tori. We study their topological properties, such as fundamental groups and equivariant cohomology algebras. Then, we discuss when the torus equivariant cohomology algebra distinguishes them up to weakly equivariant homeomorphism.
In the present paper, we characterize Fano Bott manifolds up to diffeomorphism in terms of three operations on matrix. More precisely, we prove that given two Fano Bott manifolds $X$ and $X'$, the following conditions are equivalent: (1) the upper triangular matrix associated to $X$ can be transformed into that of $X'$ by those three operations; (2) $X$ and $X'$ are diffeomorphic; (3) the integral cohomology rings of $X$ and $X'$ are isomorphic as graded rings. As a consequence, we affirmatively answer the cohomological rigidity problem for Fano Bott manifolds.
Rigidity problems in toric topology, a survey
在\cite{ma-su08}中讨论了复曲面拓扑中的几个刚性问题。在本文中，我们综述了这些问题的结果，包括最近的发展。
Classification of complex projective towers up to dimension 8 and cohomological rigidity
复射影塔或简称为$\mathbb CP$ -塔是从一点开始的迭代复射影纤维化。在本文中，我们将所有的6维$\mathbb CP$ -塔分类到同构，并且作为结果，我们证明了所有这样的流形是上同调刚性的，即，它们由它们的上同调环在同构下完全确定。我们还表明，上同调刚性是无效的8维$\mathbb CP$ -塔分类$\mathbb CP^1$ -纤维化$\mathbb CP^3$的单同态。作为推论，我们证明了这样的$\mathbb CP$ -塔是同伦等价的.
Equivariant bordism rigidity for toric manifolds
本文提出等变配边方法，证明环面流形同构等价于弱等变酉配边，并给出全定向拟环面流形的刚性结果，以拓扑框架检测几何与组合刚性。
Quoric manifolds are the quaternionic analogue of toric manifolds. They admit a locally nice action of $(S^3)^n$ and the quotient is a manifold with corners. We show that they satisfy equivariant rigidity. More precisely, any locally linear $(S^3)^n$-manifold that it is equivariantly homotopic equivalent to a quoric manifold is equivariantly homeomorphic to it. The proof is given by generalising the methods of used in Coxeter and toric manifolds.
Cohomological rigidity of real Bott manifolds
实Bott流形是从一点开始的迭代RP ^1丛的全空间，其中每个RP ^1丛是两个实线丛的Whitney和的投影化。证明了两个实Bott流形是同构的，如果它们的系数为Z/2的上同调环是同构的.一个实Bott流形是一个实环面流形，并且在初等交换2-群的自然作用下有一个平坦的黎曼度量不变量。我们还证明了相反的情况也是成立的，即在初等交换2-群的作用下存在平坦黎曼度量不变量的实环面流形是实Bott流形。
Equivariant Elliptic Cohomology and Rigidity
复系数等变椭圆上同调由Ginzburg，Kapranov和Vasserot公理化地定义，并由Grojnowski构造。给出了S^1-等变椭圆上同调的一个不变定义，并利用它给出了Witten关于椭圆亏格的刚性定理的一个完全上同调证明。我们还陈述并证明了椭圆族的一个刚性定理。
Cohomological rigidity of manifolds with torus actions: I
本文针对 moment-angle 流形与拓扑环面流形，证明满足组合条件的单纯球面所对应流形具有上同调刚性，并通过面环的 Tor-代数确定其组合类型。
We introduce and study several new topological operads that should be regarded as nonsymmetric analogues of the operads of little 2-disks, framed little 2-disks, and Deligne-Mumford compactifications of moduli spaces of genus zero curves with marked points. These operads exhibit all the remarkable algebraic and geometric features that their classical analogues possess; in particular, it is possible to define a noncommutative analogue of the notion of cohomological field theory with similar Givental-type symmetries. This relies on rich geometry of the analogues of the Deligne-Mumford spaces, co…
Cohomological rigidity and the number of homeomorphism types for small covers over prisms
本文在M. W. Hirsch（1976）\cite [Chapter 8，§2 Gluing Manifolds Together]{h}，我们给出了构造简单凸多胞体上两个小覆盖之间的同胚的一种方法。因此，我们用$m\geq 3$对三维棱镜$P^3(m)$上的所有小覆盖进行分类，直到同胚。本文引入了两个色棱镜不变量和两个小覆盖的具有${\Bbb Z}_2$ -系数的普通上同调环不变量。在大多数情况下，这些不变量可以构成棱柱上所有小覆盖的同胚类型的完整不变量系统。然后，我们证明了上同调刚性对棱柱$P^3(m)$上的所有小覆盖都成立（即，具有${\Bbb Z}_2$的上同调环-$P^3(m)$上的所有小覆盖的系数确定它们的同胚类型）。此外，我们还计算了$P^3(m)$上所有小覆盖的同胚类型数。
A family of closed manifolds is called cohomologically rigid if a cohomology ring isomorphism implies a diffeomorphism for any two manifolds in the family. We establish cohomological rigidity for large families of 3-dimensional and 6-dimensional manifolds defined by 3-dimensional polytopes. We consider the class P of 3-dimensional combinatorial simple polytopes, different from a tetrahedron, whose facets do not form 3- and 4-belts. This class includes mathematical fullerenes, i.e. simple 3-polytopes with only 5-gonal and 6-gonal facets. By a theorem of Pogorelov, any polytope from P admits a r…
Homotopy rigidity for quasitoric manifolds over a product of simplices
本文研究上同调刚性问题的变体，证明当两个拟环流流形的上同调环同构且其中一个的多胞体为单纯形乘积时，它们在局部化掉显式素数列表后同伦等价。
We prove that if $Γ$ is a lattice in the group of isometries of a symmetric space of non-compact type without euclidean factors, then the virtual cohomological dimension of $Γ$ equals its proper geometric dimension.
A dissection of a polygon is obtained by drawing diagonals such that no two diagonals intersect in their interiors. In this paper, we define a toric variety of Schröder type as a smooth toric variety associated with a polygon dissection. Toric varieties of Schröder type are Fano generalized Bott manifolds, and they are isomorphic if and only if the associated Schröder trees are the same as unordered rooted trees. We describe the cohomology ring of a toric variety of Schröder type using the associated Schröder tree and discuss the cohomological rigidity problem.
Describing toric varieties and their equivariant cohomology
拓扑上，紧环面簇可以被构造为标识空间：它们是紧环面与扇的序复形的乘积的乘积。我们给出了一个详细的证明这一事实，将其扩展到非紧的情况下，并得出几个，主要是上同调的结论。特别是，我们表明，等变积分上同调的环面品种可以描述的分段多项式的风扇，如果普通的积分上同调集中在偶数度。这将Bahri-Franz-Ray的一个结果推广到了非紧情形。我们还研究了积分上同调中的挠率现象。
Link Bundles of Compact Toric Varieties of Real Dimension 8
本文研究了8维紧致托里奇变种的链接贝蒂数，证明其非组合不变参数与相关变种的参数相等，并提供了代数描述。
A simple polytope $P$ is called $B$-rigid if its combinatorial type is determined by the cohomology ring of the moment-angle manifold $\mathcal{Z}_P$ over $P$. We show that any tensor product decomposition of this cohomology ring is geometrically realized by a product decomposition of the moment-angle manifold up to equivariant diffeomorphism. As an application, we find that $B$-rigid polytopes are closed under products, generalizing some recent results in the toric topology literature. Algebraically, our proof establishes that the Koszul homology of a Gorenstein Stanley-Reisner ring admits a …
Chern-Simons theory and cohomological invariants of representation varieties
本文证明离散群表示下拉回齐次形式的局部刚性定理，并据此说明局部建模于约化齐次空间的闭流形体积在结构形变下不变，核心在于用陈-韦伊形式与陈-西蒙斯形式重新诠释不变形式代数。
The main goal of this article is to study the cohomology rings and their applications of moment-angle complexes associated to Gorenstein* complexes, especially, the applications in combinatorial commutative algebra and combinatorics. First, we give a topological characterization of Gorenstein* complexes in terms of Alexander duality (as an application we give a topological proof of Stanley's Theorem). Next we give some cohomological transformation formulae of $\mathcal {Z}_{K}$, which are induced by some combinatorial operations on the Gorenstein* complex $K$, such as the connected sum operati…
Cohomological properties of the Vietoris--Rips Complex of a Hypercube Graph
本文通过托利拓学框架研究超立方体图的维蒂里斯-里斯复形的上同调性质，利用总支配不变量和谱方法建立连通性的下界，推翻Shukla猜想并获得全局上界，引入鬼顶点工具检测和证明上同调类的线性无关性。
On the cohomology of torus manifolds
环面流形是由具有非空不动点集和一些附加方向数据的半维环面作用的偶数维流形。它可以被认为是一个深远的概括环面流形从代数几何。环面流形的轨道空间具有丰富的组合结构，例如，它是一个带有角的歧管，只要动作是当地标准的。在这里，我们研究环面流形的上同调性质和它们的轨道向量的组合之间的关系。证明了环面流形的上同调环是由二维类生成的当且仅当其商是同调多胞形。在这种情况下，我们从复曲面几何中恢复熟悉的图像：等变上同调是神经单纯复形的面环，而普通上同调是通过分解出某些线性型而得到的。在更一般的情况下，我们证明了环面流形的奇次上同调为零当且仅当其轨道空间是面非循环的。虽然在这种情况下上同调不再是二次生成的，但等变上同调仍然同构于适当单纯偏序集的面环。
Ranks of homotopy and cohomology groups for rationally elliptic spaces and algebraic varieties
本文研究了有理椭圆空间与代数簇同伦群和上同调群的秩，证明了自乘积 Poincaré 多项式及混合 Hodge 结构生成函数值间的不等式，并引入稳定化阈值不变量探讨其与 Hilali 猜想的联系。
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