Brouwer degree, domination of manifolds, and groups presentable by products
Brouwer degree, domination of manifolds, and groups presentable by products
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布劳威尔度、流形的支配以及乘积表示的群体
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发表时间：
2016
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通讯作者：
P. Harpe
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作者：
P. Harpe
对于同维定向连通闭流形，存在一个传递关系：$M$支配$N$，或$M \ge N$，如果存在一个从$M$到$N$的非零度连续映射。第1节是一个提醒的概念程度（布劳威尔，霍普夫），第2节显示的例子，支配和第一组障碍，支配由于霍普夫，第3节描述的障碍方面格罗莫夫的单纯体积。
在第4节中，我们讨论了一个特定的问题，即一个给定的流形何时可以（或不能）被一个乘积支配。由于D. Kotschick和C. L“oh：一个群可用一个乘积表示，如果它包含两个生成一个有限指数子群的无限可交换子群。最后一节展示了一小部分无法通过产品展示的群体;例子包括适当的Coxeter群体。
For oriented connected closed manifolds of the same dimension, there is a transitive relation: $M$ dominates $N$, or $M \ge N$, if there exists a continuous map of non-zero degree from $M$ onto $N$. Section 1 is a reminder on the notion of degree (Brouwer, Hopf), Section 2 shows examples of domination and a first set of obstructions to domination due to Hopf, and Section 3 describes obstructions in terms of Gromov's simplicial volume.
In Section 4 we address the particular question of when a given manifold can (or cannot) be dominated by a product. These considerations suggest a notion for groups (fundamental groups), due to D. Kotschick and C. L\"oh: a group is presentable by a product if it contains two infinite commuting subgroups which generate a subgroup of finite index. The last section shows a small sample of groups which are not presentable by products; examples include appropriate Coxeter groups.