Twist groups of compact 3-manifolds
Twist groups of compact 3-manifolds
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紧凑型 3 歧管扭转组
DOI：
10.1016/0040-9383(85)90015-1
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发表时间：
1985
期刊：
影响因子：
--
通讯作者：
Darryl McCullough
中科院分区：
文献类型：
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作者：
Darryl McCullough
设D是3流形m中的一个适当嵌入的2圆盘。选择D的一个与D2 x I同纯的邻域N，其中N N dM对应于aD2 x I，我们定义一个关于D的扭转为同纯t，使得tD (re2 ' @, t)=(re2 ' @+ t), t) if (re2rie, t)~ N, t,(x)= x if x $ N，注意tDlaM是一个关于aD的Dehn扭转。扭转有两个方向，对应于D正则邻域的取向选择。除了这种选择外，全映射类群& (M)中的同位素类（to）只依赖于M中D的环境同位素类，所有扭转同位素类的集合产生扭转群F (M)，即JZ (M)的正规子群。很容易看出，t与单位映射1m是同伦的，因此5 (M)包含在自然同态& (M)+ Out （ni (M)）的核中，该同态取(h)到h。但是rD laM只有在aD结合到aM中的圆盘或莫比乌斯带时才与1am同位素，因此rD通常不与1am同位素。
LET D be a properly imbedded 2disc in a 3manifold M. Choosing a neighborhood N of D which is homeomorphic to D2 x I with N n dM corresponding to aD2 x I, we define a twist about D to be the homeomorphism t, such that tD (re2”@, t)=(re2ri@+ t), t) if (re2rie, t)~ N and t,(x)= x if x $ N. Note that tDlaM is a Dehn twist about aD. There are two directions for twisting, corresponding to the choices of orientation for a regular neighborhood of D. Except for this choice, the isotopy class (to) in the full mapping class group& (M) depends only on the ambient isotopy class of D in M. The collection of all isotopy classes of twists generates the twist group F (M), a normal subgroup of JZ (M).It is easy to see that t, is homotopic to the identity map 1 M and hence 5 (M) is contained in the kernel of the natural homomorphism& (M)+ Out (ni (M)) that takes (h) to h,. But rD laM is isotopic to 1 aM only when aD bounds a disc or a Mobius band in aM, hence rD is often not isotopic to 1,.