Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003;
Institute of Mathematics for Industry, Kyushu University, Fukuoka 819-0395, Japan.
Proc Natl Acad Sci U S A. 2018 Oct 23;115(43):10894-10900. doi: 10.1073/pnas.1717175115. Epub 2018 Oct 22.
Shapes of 4D spaces can be studied effectively via maps to standard surfaces. We explain, and illustrate by quintessential examples, how to simplify such generic maps on 4-manifolds topologically, to derive simple decompositions into much better-understood manifold pieces. Our methods not only allow us to produce various interesting families of examples but also to establish a correspondence between simplified broken Lefschetz fibrations and simplified trisections of closed, oriented 4-manifolds.
4D 空间的形状可以通过映射到标准曲面来有效地研究。我们通过典型的例子来解释和说明如何在拓扑上将这种通用映射简化到 4 流形上，从而得到更简单的分解成更容易理解的流形部分。我们的方法不仅使我们能够生成各种有趣的例子族，还可以在简化的断开 Lefschetz 纤维化和封闭的、定向的 4 流形的简化三分之间建立对应关系。
