四维流形上一些问题的研究
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摘要
本文围绕着四维流形上的群作用及相关问题,运用Seiberg-Witten理论、G-符号差公式以及Lefschetz不动点定理等工具,研究四维流形上的一些拓扑性质,主要包括以下几个方面：
1.四维流形上表示某些给定同调类的嵌入曲面的亏格；
2.辛椭圆曲面上同调平凡的群作用；
3.相交形式为E8(?) E8的拓扑4-流形上的素自同构.
第一章介绍了四维流形的研究背景、意义以及应用,特别介绍了近年来国内外学者在四维流形上的群作用以及相关问题的研究概况以及主要的研究成果.
第二章主要介绍了本文研究工作中所需要的一些基本概念、基础知识和预备工具.包括相交形式、分类定理和格点以及四维流形上群作用、整表示和实现定理、Spin结构、辛结构等,并介绍了Seiberg-Witten理论,以及G-符号差公式和Lefchetz不动点定理.
第三章利用Seiberg-Witten理论的结果,特别是10/8-定理,并结合古典的分支复叠的方法,对四维流形上的表示同调类的嵌入曲面进行了研究,在四维流形的一阶同调H1(X；Z)为有限群的情况下,得到表示某些满足可除性的二阶同调类的嵌入曲面的亏格的下界.
第四章利用G-符号差公式,研究了E8(?) E84-流形在Z7、Z5和Z3作用情形下的所有可能的整表示.给出了可以被伪自由的群作用实现的表示,并排除了不能由含有二维不动点集的局部线性群作用实现的表示.
第五章利用Seiberg-Witten-Taubes理论,在Chen和Kwasik研究工作的基础上,特别的利用有限多个J-全纯曲线uiCi以及辛作用下不动点集MG的结构,深入研究了辛椭圆曲面上的同调平凡群作用.一方面,对伪自由情形下辛同伦椭圆曲面E(n)上同调平凡的循环群作用进行讨论,得到了p=3时同伦椭圆曲面上不存在同调平凡的辛Zp作用,p=5时允许一个非平凡的辛同调平凡作用,并给出相应的不动点数据；另一方面,通过详细讨论p取2、3、5以及其他素数时对应在G-符号差公式中的亏值,证明了带扰动c1(K).[ω]<16的同伦极小辛椭圆曲面E(4)上保持辛结构的同调平凡的循环群作用是平凡的.
In this dissertation, around the group actions on four-manifolds and some related prob-lems, we use Seiberg-Witten theory, G-signature formula and Lefchetz fixed point theorem to discuss some problems on the four-manifolds, including the following topics:
1. The genus of surfaces representing some homology classes on four-manifolds;
2. The homologically trivial cyclic group actions on symplectic elliptic surfaces;
3. The automorphisms of the E8(?) E84-manifold with prime periods.
In Chapter1, we give a review on the background, importance and application of the four-manifold theory. We also introduce main recent achievements and open questions in the field about group actions on four-manifolds and other relative problems.
In Chapter2, basic concepts, knoweledges and tools are given, such as intersection forms, classification theorem and lattices, as well as group actions on four-manifolds, integral rep-resentations, realizing theorem, Spin structure, symplectic structure, Seiberg-Witten theory, G-signature formula and Lefchetz fixed point theorem.
In Chapter3, we use the results from Seiberg-Witten theory, especially10/8-theorem, and classical methods about branched covering, to study embedded surfaces in four-manifolds, which represent some2-homology classes. We obtain some lower bounds of the genus of some embedded surfaces representing some2-homology classes divisible by2r in certain four-manifolds with H1(X;Z) finite.
In Chapter4, we consider all the possible integral representations of the E8(?)E84-manifold under Z7、Z5and Z3actions respectively with G-signature formula. We recognize those rep-resentations that can be realizable by locally linear pseudofree actions and exclude those that cannot be realizable by locally linear actions with two-dimensional fixed components.
In Chapter5, we start from some basic Seiberg-Witten knowledge of Taubes, and focus on the question of homologically trivial actions on symplectic elliptic surfaces, using the results of Chen and Kwasik about the union UiCi of finite J-holomorphic curves and the structure of fixed point set under symplectic actions. We discuss the pseudofree Zp actions on the symplec-tic homotopy elliptic surfaces E(n) with c12=0, and obtain there are no homologically trivial symplectic Zp actions on symplectic homotopy elliptic surfaces when p=3, and when p=5, there is a nontrivial symplectic homologically trivial action with some fixed datum. Meanwhile, by disccussing defects in the G-signature formula when p=2,3,5and other primes, we prove that all the homologically trivial cyclic actions preserving symplectic structures on the minimal symplectic elliptic surfaces E(4) with a disturb c1(K)·[ω]<16are trivial.
1.四维流形上表示某些给定同调类的嵌入曲面的亏格；
2.辛椭圆曲面上同调平凡的群作用；
3.相交形式为E8(?) E8的拓扑4-流形上的素自同构.
第一章介绍了四维流形的研究背景、意义以及应用,特别介绍了近年来国内外学者在四维流形上的群作用以及相关问题的研究概况以及主要的研究成果.
第二章主要介绍了本文研究工作中所需要的一些基本概念、基础知识和预备工具.包括相交形式、分类定理和格点以及四维流形上群作用、整表示和实现定理、Spin结构、辛结构等,并介绍了Seiberg-Witten理论,以及G-符号差公式和Lefchetz不动点定理.
第三章利用Seiberg-Witten理论的结果,特别是10/8-定理,并结合古典的分支复叠的方法,对四维流形上的表示同调类的嵌入曲面进行了研究,在四维流形的一阶同调H1(X；Z)为有限群的情况下,得到表示某些满足可除性的二阶同调类的嵌入曲面的亏格的下界.
第四章利用G-符号差公式,研究了E8(?) E84-流形在Z7、Z5和Z3作用情形下的所有可能的整表示.给出了可以被伪自由的群作用实现的表示,并排除了不能由含有二维不动点集的局部线性群作用实现的表示.
第五章利用Seiberg-Witten-Taubes理论,在Chen和Kwasik研究工作的基础上,特别的利用有限多个J-全纯曲线uiCi以及辛作用下不动点集MG的结构,深入研究了辛椭圆曲面上的同调平凡群作用.一方面,对伪自由情形下辛同伦椭圆曲面E(n)上同调平凡的循环群作用进行讨论,得到了p=3时同伦椭圆曲面上不存在同调平凡的辛Zp作用,p=5时允许一个非平凡的辛同调平凡作用,并给出相应的不动点数据；另一方面,通过详细讨论p取2、3、5以及其他素数时对应在G-符号差公式中的亏值,证明了带扰动c1(K).[ω]<16的同伦极小辛椭圆曲面E(4)上保持辛结构的同调平凡的循环群作用是平凡的.
In this dissertation, around the group actions on four-manifolds and some related prob-lems, we use Seiberg-Witten theory, G-signature formula and Lefchetz fixed point theorem to discuss some problems on the four-manifolds, including the following topics:
1. The genus of surfaces representing some homology classes on four-manifolds;
2. The homologically trivial cyclic group actions on symplectic elliptic surfaces;
3. The automorphisms of the E8(?) E84-manifold with prime periods.
In Chapter1, we give a review on the background, importance and application of the four-manifold theory. We also introduce main recent achievements and open questions in the field about group actions on four-manifolds and other relative problems.
In Chapter2, basic concepts, knoweledges and tools are given, such as intersection forms, classification theorem and lattices, as well as group actions on four-manifolds, integral rep-resentations, realizing theorem, Spin structure, symplectic structure, Seiberg-Witten theory, G-signature formula and Lefchetz fixed point theorem.
In Chapter3, we use the results from Seiberg-Witten theory, especially10/8-theorem, and classical methods about branched covering, to study embedded surfaces in four-manifolds, which represent some2-homology classes. We obtain some lower bounds of the genus of some embedded surfaces representing some2-homology classes divisible by2r in certain four-manifolds with H1(X;Z) finite.
In Chapter4, we consider all the possible integral representations of the E8(?)E84-manifold under Z7、Z5and Z3actions respectively with G-signature formula. We recognize those rep-resentations that can be realizable by locally linear pseudofree actions and exclude those that cannot be realizable by locally linear actions with two-dimensional fixed components.
In Chapter5, we start from some basic Seiberg-Witten knowledge of Taubes, and focus on the question of homologically trivial actions on symplectic elliptic surfaces, using the results of Chen and Kwasik about the union UiCi of finite J-holomorphic curves and the structure of fixed point set under symplectic actions. We discuss the pseudofree Zp actions on the symplec-tic homotopy elliptic surfaces E(n) with c12=0, and obtain there are no homologically trivial symplectic Zp actions on symplectic homotopy elliptic surfaces when p=3, and when p=5, there is a nontrivial symplectic homologically trivial action with some fixed datum. Meanwhile, by disccussing defects in the G-signature formula when p=2,3,5and other primes, we prove that all the homologically trivial cyclic actions preserving symplectic structures on the minimal symplectic elliptic surfaces E(4) with a disturb c1(K)·[ω]<16are trivial.
引文
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