The space of incompressible surfaces in a 2-bridge link complement
The space of incompressible surfaces in a 2-bridge link complement
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2 桥连杆补中不可压缩表面的空间
DOI：
10.1090/s0002-9947-1988-0924770-8
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发表时间：
1988
影响因子：
1.3
通讯作者：
A. Hatcher
中科院分区：
文献类型：
--
作者：
W. Floyd;A. Hatcher
明确地计算了2-桥链补的投影层空间。本文构造了一个多面体P ε（S ~ 3-Lp/q），它的有理点以一种自然的方式与2-桥链Lp/q c S ~ 3外的不可压缩曲面的射影合痕类双射对应.这里的“投影”意味着我们通过标量乘法来分解出一个曲面的任何数量的平行副本。（不假定曲面是连接的。）我们期望P ε（S3 Lp/q）将是[5和12]中对一般紧致不可约3-流形定义和研究的S3 Lp/q 33的“射影层空间”。瑟斯顿的曲面射影层空间理论中没有出现的一个意想不到的复杂性是P ε（S3 Lp/q）经常是非紧的，例如怀特黑德环L3/8（见图5.4，左上角）。然而，在一般理论中，Pe（S3-Lp/q）有一个自然紧化P7（S3-Lp/q），它是一个有限多面体。为了构造P ε（S3-Lp/q）3，我们首先找到一个相当自然的分支曲面的有限集合Bi C S3-Lp/q，它携带S3 Lp/q中的所有不可压缩曲面。每个Bi都有一个凸胞ci，它的有理点参数化了Bi所携带的曲面的投影类。Ci的不同有理点可以确定同位素表面，然而，由于在Bi的互补中（在表面上的火车轨道的互补中的“digon”区域的类似物）推动部分表面穿过产品区域的可能性。这导致了Ci到另一个凸多面体胞元ci上的线性投影Pi：Ci Ci °f Ci，使得在ci的内部，投影合痕曲面类与Pi的纤维重合。然而，这些合痕关系可能不会持续超过ci的边界。即，传递到Ci的面对应于传递到Bi的分支子表面，并且该分支子表面的非积互补区域可以被Bi分解为Bi的积互补区域。在这种情况下，Ci的这一面的有理点不对应于不可压缩曲面的（投影）合痕类，而是对应于具有这些极限“幻影”合痕关系的不可压缩曲面。空间P7（53 Lp/q）通过以最自然的方式识别它们的面，区分相同表面集合之间的不同“幻影”同位素，从单元Ci形成。PZ（S3-Lp/q）由PS（S3 Lp/q）的开放单元组成，其中“幻影”同位素是实际同位素。编辑于1986年4月12日收到。1980年数学学科分类（1985年修订）。第57 M25
Projective lamination spaces for 2-bridge link complements are computed explicitly. In this paper we construct a polyhedron P£(S3-Lp/q) whose rational points correspond bijectively, in a natural way, with the projective isotopy classes of incompressible surfaces in the exterior of a 2-bridge link Lp/q c S3. Here "projective" means that we factor out by scalar multiplication taking any number of parallel copies of a surface. (Surfaces are not assumed to be connected.) We expect that P£(S3 Lp/q) will turn out to be the "projective lamination space of S3 Lp/q33 as defined and studied for general compact irreducible 3-manifolds in [5 and 12]. An unexpected complication not present in Thurston's theory of projective lamination spaces for surfaces is the fact that P£(S3 Lp/q) is frequently noncompact, for example for the Whitehead link L3/8 (see Figure 5.4, upper left-hand corner). However, as in the general theory, P£(S3 Lp/q) has a natural compactification P7(53-Lp/q) which is a finite polyhedron. To construct P£(S3-Lp/q)3 we first find a fairly natural finite collection of branched surfaces Bi C S3-Lp/q which carry all the incompressible surfaces in S3Lp/q. To each Bi is associated a convex cell ci whose rational points parametrize the projective classes of surfaces carried by Bi. DiSerent rational points of Ci can determine isotopic surfaces, however, due to the possibility of pushing parts of surfaces across product regions in the complement of Bi (the analogue of "digon" regions in the complement of a train track on a surface). This leads to a linear projection Pi: Ci Ci °f Ci onto another convex polyhedral cell ci, such that over the interior of ci, projective isotopy classes of surfaces coincide with fibers of Pi. However, these isotopy relations may not persist over the boundary of ci. Namely, passing to a face of Ci corresponds to passing to a branched subsurface of Bi, and a nonproduct complementary region of this branched subsurface may be decomposed by Bi into product complementary regions of Bi. In this case, rational points of this face of Ci correspond not to (projective) isotopy classes of incompressible surfaces, but to incompressible surfaces with these limiting "phantom" isotopy relations. The space P7(53 Lp/q) is foed from the cells Ci by identifying their faces in the most natural way, distinguishing different "phantom" isotopies between the same sets of surfaces. PZ (S3-Lp/q) consists of the open cells of PS (S3 Lp/q) for which the "phantom" isotopies are actual isotopies. Received by the editors April 12, 1986. 1980 Mathematics Subject Classification (1985 Revtaion). Primary 57M25.
沿 A Campo 分裂结进行有限 Dehn 手术
DOI：
--
发表时间：
2006
期刊：
Advanced Studies in Pure Mathematics 43 43
影响因子：
--
作者：
T.KADOKAMI;Y.YAMADA;Yuichi YAMADA;山田 裕一;山田 裕一;T.KADOKAMI and Y.YAMADA;Yuichi YAMADA
通讯作者：
Yuichi YAMADA