AI 中文总结
本文通过混合纤维和构造辛四流形，分类了欧拉数4且符号数0的情形，并利用辛平移产生无穷多个不同胚例子，其有理上同调环为 $S^2\times S^2$ 或 $\mathbb CP^2\\#\overline{\mathbb CP}^{\\,2}$。
AI 中文摘要
我们通过沿直纹曲面中的显式循环多截面进行混合纤维和来构造辛四流形。对于 $\Sigma_g\times S^2$ 中一个连通的无分支 $p$ 次多截面，我们确定了其补空间的第一同调和基本群，并证明其边界是不可压缩的。由此可知，两个这样的补空间不能直接粘合得到单连通流形；此外，每个直接和的第二同调都保留了由覆盖次数决定的有限商群。我们对具有欧拉示性数 $4$ 和符号数 $0$ 的混合和进行了分类。在交换两个被加数的意义下，恰好存在三种可能性，分别对应于次数对 $(2,3)$、$(2,4)$ 和 $(3,3)$。对于这些情形中的每一种，适当的适配乘积框架辛粘合都具有 $S^2\times S^2$ 的有理上同调环。通过辛平移变换改变粘合方式，可以产生无穷多个两两不同胚的例子，这些例子由它们有限第一同调群的无界阶数来区分。我们还构造了 $(2,4)$ 情形的扭曲直纹曲面类比。显式的有限全纯多截面在 $\Sigma_3$ 和 $\Sigma_2$ 上的非平凡 $S^2$ 丛中分别给出了类 $2S_3-F_3$ 和 $4S_2-2F_2$ 中的连通平方零辛曲面。更一般地，对于非平凡丛中的平方零 $p$ 次多截面，其补空间的第一同调为 $\mathbb Z^{2g}\oplus\mathbb Z/(p/2)$。在扭曲 $(2,4)$ 情形中，适当的粘合具有 $b_1=0$、$b_2=2$ 和零符号数，并且每个这样的和都是非自旋的。因此它们具有 $\mathbb CP^2\\#\overline{\mathbb CP}^{\\,2}$ 的有理上同调环。我们将这些构造与作者 2006 年通过纽结手术和扭曲纤维和构造的具有 $S^2\times S^2$ 整系数上同调的极小辛四流形进行了比较。
英文摘要
We construct symplectic four-manifolds by taking mixed fiber sums along explicit cyclic multisections in ruled surfaces. For a connected unbranched degree-$p$ multisection in $Σ_g\times S^2$, we determine the first homology and fundamental group of the complement and prove that its boundary is incompressible. It follows that no direct gluing of two such complements can be simply connected; moreover, the first homology of every direct sum retains finite quotients determined by the covering degrees. We classify the mixed sums having Euler characteristic $4$ and signature $0$. Up to interchanging the two summands, exactly three possibilities occur, corresponding to the degree pairs $(2,3)$, $(2,4)$, and $(3,3)$. For each of these cases, suitable adapted product-framed symplectic gluings have the rational cohomology ring of $S^2\times S^2$. Varying the gluing by symplectic transvections produces infinitely many pairwise nondiffeomorphic examples, distinguished by the unbounded orders of their finite first homology groups. We also construct the twisted ruled analogue of the $(2,4)$ case. Explicit finite-holonomy multisections give connected square-zero symplectic surfaces in the classes $2S_3-F_3$ and $4S_2-2F_2$ in the nontrivial $S^2$-bundles over $Σ_3$ and $Σ_2$, respectively. More generally, for a square-zero degree-$p$ multisection in the nontrivial bundle the complement has first homology $\mathbb Z^{2g}\oplus\mathbb Z/(p/2)$. Suitable gluings in the twisted $(2,4)$ case have $b_1=0$, $b_2=2$, and signature zero, and every such sum is non-spin. Hence they have the rational cohomology ring of $\mathbb CP^2\#\overline{\mathbb CP}^{\,2}$. We compare these constructions with the author's 2006 construction of minimal symplectic four-manifolds having the integral cohomology $S^2\times S^2$, obtained via knot surgery and twisted fiber sums.
Comments23 pages, 3 figures
