Classification of real Bott manifolds and acyclic digraphs
Classification of real Bott manifolds and acyclic digraphs
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DOI：
10.1090/tran/6896
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发表时间：
2010-06
影响因子：
1.3
通讯作者：
Suyoung Choi;M. Masuda;Sang-il Oum
中科院分区：
文献类型：
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作者：
Suyoung Choi;M. Masuda;Sang-il Oum
我们完全刻画了真实的Bott流形的三个简单的矩阵运算的平方二进制矩阵的严格上三角矩阵的行和列同时置换。证明了具有Z/2系数的真实的Bott流形的上同调环之间的分次环同构是由真实的Bott流形之间的仿射环同构诱导的.这种特征也可以用有向无环图上的图操作来组合描述。使用这种组合的解释，我们证明了分解的真实的Bott流形到产品的不可分解的真实的Bott流形是唯一的排列的不可分解的因素。最后，我们从图论的角度给出了真实的Bott流形的一些数值不变量，并讨论了它们的拓扑意义。作为一个副产品，我们证明了toral秩猜想成立的真实的Bott流形。
We completely characterize real Bott manifolds up to diffeomorphism in terms of three simple matrix operations on square binary matrices obtained from strictly upper triangular matrices by permuting rows and columns simultaneously. We also prove that any graded ring isomorphism between the cohomology rings of real Bott manifolds with Z/2 coefficients is induced by an affine diffeomorphism between the real Bott manifolds. This characterization can also be described combinatorially in terms of graph operations on directed acyclic graphs. Using this combinatorial interpretation, we prove that the decomposition of a real Bott manifold into a product of indecomposable real Bott manifolds is unique up to permutations of the indecomposable factors. Finally, we produce some numerical invariants of real Bott manifolds from the viewpoint of graph theory and discuss their topological meaning. As a by-product, we prove that the toral rank conjecture holds for real Bott manifolds.