Automorphisms of a 3-dimensional handlebody
Automorphisms of a 3-dimensional handlebody
复制标题
3 维手柄的自同构
DOI：
10.1007/bf02992786
复制
发表时间：
1964
期刊：
影响因子：
--
通讯作者：
H. B. Griffiths
中科院分区：
文献类型：
--
作者：
H. B. Griffiths
所谓三维球体T~，我们指的是一个有n个实心柄的实心球。在欧氏空间R3中可以构造T~的各种简单同胚，它们揭示了Tn的某些对称性.本文研究了其中较明显的几种，计算了它们对T~及其边界面~ T~的基本群：h（T~），：h（8 T~）的影响。其中一些在zr 1（T~）上诱导恒等式，但在zrl（~ T~）上不诱导恒等式，虽然我们不知道这些恒等式的列表在多大程度上是完整的，但我们得到（Th. 10. 1)rl（OT~）的自同构由T~的拓扑自同构诱导的一个充要条件.现在，NIELSEN [3]的一个经典结果表明，zh（OT~）的每个自同构都是由O Tn的一个自同构诱导的，而HI（~ Tn）的同调模拟当然是错误的，因为相交数必须保持。我们利用T~的某些对称性证明了相应的结果（3.1，3.3），即：r1（T~）或HI（T，~）的每一个自同构都是由T~的一个自同构诱导的;我们只是简单地表明，所有的，Tietze变换（它们生成zrl（T~）的自同构群）是由这些对称性诱导的。在这项工作的手稿完成后，在ZIESC~ O的文[5]中，我们的定理3.1（关于：h（T~））用不同的方法得到了证明。第238页，Satz 1）。我们已经使用了一个结果，从该文件，以简化我们的证明Th。10.1（上面提到的）。
By a 3-dimensional handlebody T~, we mean a solid ball with n solid handles. Various simple homeomorphs of T~ can be constructed in Euclidean space R 3, and these reveal certain symmetries of Tn. The more obvious ones of these are studied in this paper, and we compute their effects on the fundamental groups: h (T~),: h (8 T~) of T~ and its boundary surface~ T~. Some of them induce the identity on zr 1 (T~) but not on zrl (~ T~), and while we do not know to what extent our list of these is complete, we obtain (Th. 10. 1) a necessary and sufficient condition for an automorphism of: rl (OT~) to be induced by a topological automorphism of T~. Now, a classical result of NIELSEN [3] states that every automorphism of zh (OT~) is induced by an automorphism of O Tn, while of course the homology analogue for HI (~ Tn) is false, since intersection numbers must be preserved. We use some of the symmetries of T~, mentioned above, to prove corresponding results (3.1, 3.3) for T~, namely that every automorphism of: r1 (T~), or of HI (T,~), is induced by an automorphism of T~; we simply show that all the,, Tietze transformations'(which generate the automorphism group of zrl (T~)) are induced by these symmetries.After a manuscript of this work was completed, there appeared the paper [5] of ZIESC~ O, in which our Theorem 3.1 (about: h (T~)) is proved by different methods (op. cit. p. 238, Satz 1). We have used one result from that paper to simplify our proof of Th. 10.1 (mentioned above).