Simplifying indefinite fibrations on 4-manifolds
Simplifying indefinite fibrations on 4-manifolds
复制标题
DOI：
10.1090/tran/8325
复制
发表时间：
2017-05
影响因子：
1.3
通讯作者：
R. Baykur;O. Saeki
中科院分区：
文献类型：
--
作者：
R. Baykur;O. Saeki
我们给出了4-流形上不定纤维化拓扑的显式简化算法，其中包括断裂Lefschetz纤维化和不定莫尔斯2-函数.该算法由移动序列，修改不确定的纤维化光滑的1-参数家庭。特别是，给定一个任意破碎的Lefschetz纤维化，我们展示了如何把它变成一个定向和嵌入轮（不定倍）的图像，一个与所有的纤维和圆轨迹连接。我们还展示了如何实现任何给定的零同源1维子流形规定的局部模型，其组件的圆形轨迹，这样一个破碎的莱夫谢茨纤维化。这些算法使我们能够给出一般4-流形上简化的断裂Lefschetz纤维化和莫尔斯2-函数的存在性的纯拓扑和构造性证明，以及近辛4-流形上具有有向嵌入圆像的断裂Lefschetz束的存在性的Auroux-Donaldson-Katzarkov定理.此外，我们还建立了4-流形的破断Lefschetz纤维化与Gay-Kirby三分性之间的对应关系，并证明了所有4-流形上的简化三分性的存在性。在此基础上，我们给出了几种新的三分性的构造，包括具有同伦不等价全空间的亏格为3的三分性的无限族，以及复有理曲面同胚类中的4-流形的奇异同亏格三分性。
We present explicit algorithms for simplifying the topology of indefinite fibrations on 4-manifolds, which include broken Lefschetz fibrations and indefinite Morse 2-functions. The algorithms consist of sequences of moves, which modify indefinite fibrations in smooth 1-parameter families. In particular, given an arbitrary broken Lefschetz fibration, we show how to turn it to one with directed and embedded round (indefinite fold) image, and to one with all the fibers and the round locus connected. We also show how to realize any given null-homologous 1-dimensional submanifold with prescribed local models for its components as the round locus of such a broken Lefschetz fibration. These algorithms allow us to give purely topological and constructive proofs of the existence of simplified broken Lefschetz fibrations and Morse 2-functions on general 4-manifolds, and a theorem of Auroux-Donaldson-Katzarkov on the existence of broken Lefschetz pencils with directed embedded round image on near-symplectic 4-manifolds. We moreover establish a correspondence between broken Lefschetz fibrations and Gay-Kirby trisections of 4-manifolds, and show the existence of simplified trisections on all 4-manifolds. Building on this correspondence, we provide several new constructions of trisections, including infinite families of genus-3 trisections with homotopy inequivalent total spaces, and exotic same genera trisections of 4-manifolds in the homeomorphism classes of complex rational surfaces.