Transverse foliations of Seifert bundles and self homeomorphism of the circle
Transverse foliations of Seifert bundles and self homeomorphism of the circle
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Seifert丛的横向叶理和圆的自同胚
DOI：
10.1007/bf02566232
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发表时间：
1981
影响因子：
0.9
通讯作者：
W. Neumann
中科院分区：
文献类型：
--
作者：
D. Eisenbud;U. Hirsch;W. Neumann
本文给出了紧曲面上的Seifert圆丛允许叶都横截于纤维的叶理的判别准则，并讨论了哪些叶理可以变形为这类叶理。在第3节中提出的横叶理的判据是涉及纤维化的塞弗特对和基底的埃塞尔数的简单数值不等式（定理3.1至3.4）。他们推广了Milnor [Mi]和Wood [W]的标准，他们将局部平凡的圆丛与可定向的全空间（也见Sullivan [Su]对Milnor和Wood的高维推广）。它们是完备的，除了基是S2的情况，在这种情况下我们只有部分结果（定理3.3，5.3）。我们的准则对CO叶理和解析叶理都是有效的，如我们在第4节中所示，因此也适用于任何中间光滑度。我们减少几何问题的存在叶状代数的方式类似于米尔诺和木材。我们设~是自同胚[：R--> R的群，它们是圆的自同胚的提升。包含群~+={f：R~R [f单调递增且f（r +1）= f（r）+1，对所有r~R}作为指标为2的子群（"翻转" t（r）=-r是9+在9中的非平凡陪集的陪集表示）。对于每一个真实的数3 '，我们写sh（3 "）~~+表示"移位" %，即sh（3"）：r，--~，r +3 "表示r~R。证明了Seifert纤维流形M的横向叶化问题等价于求M的一个同态7rl（M）~的问题，该同态取M的一个非奇异纤维类到元素sh（1）~+（定理3.5）。这反过来又相当于一个问题的代表某一产品的共轭的转变为产品的一定数量的recruitors * 作者希望表达他们的感激之情，以SFB 40和数学研究所的波恩大学，在其屋顶下的大部分工作，本文所做的。第一个和最后一个作者也感谢NSF的部分支持。
In this paper we give criteria for a Seifert circle bundle over a compact surface to admit foliations whose leaves are all transverse to the fibers, and we discuss which foliations may be deformed to foliations of this type. Our criteria for transverse foliations, presented in Section 3, are simple numerical inequalities involving the Seifert pairs of the fibration and the euler number of the base (Theorems 3.1 to 3.4). They generalize criteria of Milnor [Mi] and Wood [W], who treat the case of locally trivial circle bundles with orientable total space (see also Sullivan [Su] for a higher dimensional generalization of Milnor and Wood). They are complete except for the case that the base is S 2, in which case we only have partial results (Theorems 3.3, 5.3). Our criteria are valid both for the case of CO foliations and, as we show in Section 4, for analytic foliations, hence also for any intermediate degree of smoothness. We reduce the geometric question of the existence of foliations to algebra in a way similar to that of Milnor and Wood. We let~ be the group of selfhomeomorphisms [: R--> R which are lifts of self-homeomorphisms of the circle. contains the group~+={f: R~ R [f monotonically increasing and f (r+ 1)= f (r)+ 1 for all r~ R} as a subgroup of index 2 (the" flip" t (r)=-r is a coset representative for the non-trivial coset of 9+ in 9). For each real number 3'we write sh (3")~~+ for the" shift" by% that is sh (3"): r,--~, r+ 3" for r~ R. It turns out that the problem of transversely foliating a Seifert fibered manifold M is equivalent to the problem of finding a homomorphism 7rl (M)~ which takes the class of a nonsingular fiber of M to the element sh (1)~+(Theorem 3.5). This in turn is equivalent to a problem of representating a certain product of conjugates of shifts as a product of a certain number of commutators in* The authors wish to express their gratitude to the SFB 40 and the mathematical institute of the University of Bonn, under whose roof most of the work of this paper was done. The first and last authors are also grateful for the partial support of the NSF.