Small Covers and the Equivariant Bordism Classification of 2-torus Manifolds
Small Covers and the Equivariant Bordism Classification of 2-torus Manifolds
复制标题
小覆盖和2环流形的等变Bordism分类
DOI：
10.1093/imrn/rnt183
复制
发表时间：
2010-08
影响因子：
1
通讯作者：
Tan Qiangbo
中科院分区：
文献类型：
--
作者：
Lü Zhi;Tan Qiangbo
本文结合Davis-Januszkiewicz小覆盖理论，从等变边形的角度讨论了2-环面流形的理论。我们在 Conner 和 Floyd 引入的无向 $G_n$ 表示代数的“对偶”代数上定义一个微分算子，其中 $G_n=(\Z_2)^n$。借助 $G_n$ 彩色图（或 mod 2 GKM 图），我们可以使用该微分算子对 2-环面流形设置下的 tom Dieck-Kosniowski-Stong 局域定理给出非常简单的描述。然后，我们应用它来研究 $n$ 维 2 环流形的 $G_n$ 等变无向棱柱分类。我们证明每个$n$维2-环面流形的$G_n$等变无向棱柱类包含一个$n$维小覆盖作为其代表，解决了[19]中提出的猜想。此外，我们还得出，所有可能维数的 2-环面流形的等变无向棱柱类形成的分级非交换环是由所有广义实 Bott 流形的类生成的（作为单纯形乘积的特殊小覆盖）。这在 $G_n$ 等变边界群或环的计算与小覆盖的 Davis-Januszkiewicz 理论之间建立了紧密的联系。作为计算应用，借助计算机，我们完全确定了所有4维2环流形的等变波棱类所形成的群的结构。最后，我们给出了2-环面流形、着色多项式、着色简单凸多面体、着色图之间的一些本质关系。
Associated with the Davis-Januszkiewicz theory of small covers, this paper deals with the theory of 2-torus manifolds from the viewpoint of equivariant bordism. We define a differential operator on the "dual" algebra of the unoriented $G_n$-representation algebra introduced by Conner and Floyd, where $G_n=(\Z_2)^n$. With the help of $G_n$-colored graphs (or mod 2 GKM graphs), we may use this differential operator to give a very simple description of tom Dieck-Kosniowski-Stong localization theorem in the setting of 2-torus manifolds. We then apply this to study the $G_n$-equivariant unoriented bordism classification of $n$-dimensional 2-torus manifolds. We show that the $G_n$-equivariant unoriented bordism class of each $n$-dimensional 2-torus manifold contains an $n$-dimensional small cover as its representative, solving the conjecture posed in [19]. In addition, we also obtain that the graded noncommutative ring formed by the equivariant unoriented bordism classes of 2-torus manifolds of all possible dimensions is generated by the classes of all generalized real Bott manifolds (as special small covers over the products of simplices). This gives a strong connection between the computation of $G_n$-equivariant bordism groups or ring and the Davis-Januszkiewicz theory of small covers. As a computational application, with the help of computer, we completely determine the structure of the group formed by equivariant bordism classes of all 4-dimensional 2-torus manifolds. Finally, we give some essential relationships among 2-torus manifolds, coloring polynomials, colored simple convex polytopes, colored graphs.
登录
查看更多内容
影响因子：
2.5
作者：
C. Ciliberto;R. Miranda
通讯作者：
C. Ciliberto;R. Miranda
2-环面流形的等变分类
DOI：
10.4064/cm115-2-3
发表时间：
2008-02
期刊：
Colloq. Math
影响因子：
--
作者：
Lu, Zhi;Masuda, Mikiya
通讯作者：
Masuda, Mikiya
影响因子：
1.3
作者：
Suyoung Choi;M. Masuda;Sang-il Oum
通讯作者：
Suyoung Choi;M. Masuda;Sang-il Oum
影响因子：
2.5
作者：
Michael W. Davis;T. Januszkiewicz
通讯作者：
Michael W. Davis;T. Januszkiewicz
影响因子：
0.9
作者：
V. M. Bukhshtaber;N. Ray
通讯作者：
V. M. Bukhshtaber;N. Ray