论文标题
在不可方向的4个manifolds上的Lefschetz纤维
Lefschetz fibrations on nonorientable 4-manifolds
论文作者
论文摘要
让$ w $是不可取向的$ 4 $二维句柄，而没有$ 3 $ - 和$ 4 $ - handles。我们表明，$ w $在$ 2 $ disk上承认了Lefschetz振动，其常规纤维是具有非空边界的不可取向的表面。这是在可定向情况下获得的Harer结果的类似物。作为推论，我们获得了$ 4 $维的证据，证明每个不可定向的$ 3 $ - manifold都承认了一个开放式书籍分解，这首先是由Berstein和Edmonds使用分支封面证明的。此外，我们以给定的$ 3 $ manifold获得的开放书的单型属于该页面的映射类组的扭曲子组。特别是，我们构建了一本明确的最小开放书籍，用于与真实投影平面的圆乘积的任意副本的连接总和。 我们还基于我们构建的不可方向的Lefschetz纤维化，获得了$ W $的相对三角图，类似于Castro首先研究的可定向案例。作为推论，我们获得了一些封闭的$ 4 $ manifolds的Trisection图，例如$ 2 $ sphere与真实投影平面的产品，通过加倍$ w $。此外，如果$ x $是一个封闭的不可取向的$ 4 $ - manifold，它承认了lefschetz纤维纤维上的$ 2 $ -SPHERE，配备了Square $ \ pm 1 $的一部分，那么我们构建了$ x $的三角图，这取决于Lefschetz纤维的消失循环确定的。最后，我们包括一些关于封闭的不可取向$ 4 $ manifolds上低生物Lefschetz纤维的简单观察结果。
Let $W$ be a nonorientable $4$-dimensional handlebody without $3$- and $4$-handles. We show that $W$ admits a Lefschetz fibration over the $2$-disk, whose regular fiber is a nonorientable surface with nonempty boundary. This is an analogue of a result of Harer obtained in the orientable case. As a corollary, we obtain a $4$-dimensional proof of the fact that every nonorientable closed $3$-manifold admits an open book decomposition, which was first proved by Berstein and Edmonds using branched coverings. Moreover, the monodromy of the open book we obtain for a given $3$-manifold belongs to the twist subgroup of the mapping class group of the page. In particular, we construct an explicit minimal open book for the connected sum of arbitrarily many copies of the product of the circle with the real projective plane. We also obtain a relative trisection diagram for $W$, based on the nonorientable Lefschetz fibration we construct, similar to the orientable case first studied by Castro. As a corollary, we get trisection diagrams for some closed $4$-manifolds, e.g. the product of the $2$-sphere with the real projective plane, by doubling $W$. Moreover, if $X$ is a closed nonorientable $4$-manifold which admits a Lefschetz fibration over the $2$-sphere, equipped with a section of square $\pm 1$, then we construct a trisection diagram of $X$, which is determined by the vanishing cycles of the Lefschetz fibration. Finally, we include some simple observations about low-genus Lefschetz fibrations on closed nonorientable $4$-manifolds.
