On 3-manifolds having surface bundles as branched coverings
On 3-manifolds having surface bundles as branched coverings
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在具有表面束作为分支覆盖层的 3-流形上
DOI：
10.1090/s0002-9939-1987-0908668-1
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发表时间：
1987
期刊：
影响因子：
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通讯作者：
J. Montesinos
中科院分区：
文献类型：
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作者：
J. Montesinos
我们给出了Sakuma的结果的一个不同的证明：每个闭的定向3-流形M有一个2-重分支覆盖空间N，它是Si上的一个面丛。我们还给出了布鲁克斯关于N可作双曲的结果的一个新的证明。我们给出了不可约3流形的例子，这些流形可以表示为S3的2 m重循环分支覆盖，对于一些不同的m，只要我们喜欢。1.在[S]中，Sakuma证明了对每一个闭的、定向的、连通的3-流形M3，存在S1，W3上的Fg-丛，其中Fg是亏格g的闭的、定向的、连通的曲面，使得W3是M3的2-重分支覆盖。他通过将亏格为g的一个单形体Xg看作f：Fg ->> Pg的映射圆柱（在图1中定义）来说明这一点。如果M有一个Heegaard分裂M = Xg U X 'g，则存在M在dPg U dPg上分支的2重覆盖，它可以通过沿沿着Pg U P' g分裂M并将所得的F9丛的两个副本粘贴在[0，1]上来构造。这个二重覆盖是W3。2.本文给出了同一结果的另一种证明。也就是说，我们显示LEMMA 1。设M3是具有开卷结构的闭定向三维流形，M3 = M（Fg> h ′，），其中Fg^是亏格为g且具有h个边界分支的紧致连通定向曲面，且其中tf>：Fg^ -∈ Fg.rt（monodromy映射）是限制于F_9 ~ th边界上的恒等映射的同胚，则存在S_1，M上的F_k-丛（4>ff（…>~1），它是M（F9 th; 4>）在2 h-分支链上分支的二重覆盖，其中k = 2g + h - 1.由于每个M3是一本开卷书M（F9，i;0）[GA，M]，我们推出推论2。每一个闭的定向连通3-流形M3包含一个2-分支链L，使得存在M3在L上分支的2-重覆盖，它是S1上的Fk-丛。莱姆的证明设2Fg，h是Fg，h的二重化，设i：2Fg，h -> 2F 9 th是交换2F 9 ih 1985年6月11日和1986年7月28日修订形式的两个副本的自然对合。1980年数学学科分类（1985年修订）。小学57 M12;中学57 M25。
We give a different proof of the result of Sakuma that every closed, oriented 3-manifold M has a 2-fold branched covering space N which is a surface bundle over Si. We also give a new proof of the result of Brooks that N can be made hyperbolic. We give examples of irreducible 3-manifolds which can be represented as 2m-fold cyclic branched coverings of S3 for a number of different m's as big as we like. 1. In [S] Sakuma proves that for every closed, oriented, connected 3-manifold M3, there exists an Fg-bundle over S1, W3, where Fg is a closed, oriented and connected surface of genus g, such that W3 is a 2-fold branched covering of M3. He shows this by thinking of a handlebody Xg, of genus g, as the mapping cylinder of /: Fg —» Pg (defined in Figure 1). If M has a Heegaard splitting M = Xg U X'g, there is a 2-fold covering of M branched over dPg U dPg that can be constructed by splitting M along Pg U P'g and pasting together two copies of the resulting F9-bundle over [0,1]. This 2-fold covering is W3. 2. In this note we give an alternative proof of the same result. Namely, we show LEMMA 1. Let M3 be a closed, oriented 3-manifold having an open book structure, M3 = M(Fg>h',), where Fg^ is a compact, connected, and oriented surface of genus g with h boundary components, and where tf>: Fg^ —► Fg.rt (the monodromy map) is a homeomorphism which restricts to the identity map on the boundary of F9thThen there exists an Fk-bundle over S1, M(4>ff(¡>~1), which is a 2-fold covering of M(F9th; 4>) branched over a 2h-component link, and where k = 2g + h — 1. Since every M3 is an open book M(F9,i;0) [GA, M] we deduce COROLLARY 2. Every closed, oriented connected 3-manifold M3 contains a 2component link L, such that there is a 2-fold covering of M3 branched over L which is an Fk-bundle over S1. PROOF OF LEMMA l. Let 2Fg,h be the double of Fg,h and let i: 2Fg,h -> 2F9th be the natural involution interchanging the two copies of 2F9ihReceived by the editors June 11, 1985, and, in revised form, July 28, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 57M12; Secondary 57M25.