【作者】 李红霞；

【导师】 刘西民；

【作者基本信息】 大连理工大学 ， 基础数学， 2007， 博士

【摘要】 本文运用Seiberg-Witten理论和等变K-理论、G-index定理（G-signature公式、G-Spin定理）以及Lefschetz不动点公式等工具，深入研究了四维流形上的有限群作用，研究主要包括以下内容：1．四维流形上的光滑有限群作用；2．椭圆曲面上的局部线性伪自由作用。第一章介绍了Seiberg-Witten理论及其应用，同时介绍了国内外学者在Seiberg-Witten理论的研究及应用中所取得的主要成果。第二章给出了本研究工作所需要的一些预备知识，主要介绍了Seiberg-Witten不变量的基本理论，并介绍了Seiberg-Witten理论的有限维逼近技巧、流形上的有限群作用、群表示论以及等变K-理论等基础知识。第三章运用Seiberg-Witten理论的有限维逼近技巧和等变K-理论等工具，研究了Spin 4-流形X上分别存在光滑交错群A₅、循环群Z₆和奇型三阶对称群S₃作用时，X的拓扑限制问题，在群作用下改进了Furuta的10／8定理。特别地，在Spin 4-流形具有非交换群A₅作用时，我们得到了：若X为光滑的具有非正符号差的Spin 4-流形，b₁（X）=0，令k=-σ（X）／16，m=b₂⁺（X），如果X上具有Spin交错群A₅作用，且满足b₂⁺（X／<s>）+b₂⁺（X／<t>）≠2b₂⁺（X／A₅），b₂⁺（X／<s>）≠0，则2k+3≤m，其中<s>和<t>分别为s=（abcde）∈A₅和t=（abc）∈A₅生成的A₅的子群。此外，我们还得到了K-理论度以及等变Dirac算子的G-指标的简单表示，最后讨论了交错群A₅在K3曲面上作用的具体例子。本章还研究了同伦等价于S²×S²的Spin 4-流形上的交错群A₅作用。在某些条件下，我们将G等变Dirac算子的G-指标的表示公式化简为Ind_A5D_X=a(1-2_ρ1+ρ4)+b（ρ₂-ρ₁），其中a，b为整数。此外，运用类似的方法我们还研究了n个S²×S²的连通和上的交错群A₅作用。第四章运用G-signature公式、G-Spin定理和Lefschetz不动点公式等工具对椭圆曲面上的局部线性伪自由Z₃作用的给出了完全的拓扑分类，我们证明：椭圆曲面E（4）上的局部线性伪自由Z₃作用共有十种类型，其中有九种类型能够真正地由椭圆曲面上的局部线性伪自由Z₃作用所实现，并且给出实现定理。此外，我们运用Seiberg-Witten不变量的mod p消失定理证明了不能在标准光滑椭圆曲面E（4）上光滑实现的局部线性伪自由Z₃作用的存在性。更多还原

【Abstract】 In this dissertation, by using the Seiberg-Witten theory, equivariant K-theory, G-indextheorem（G-signature formula, G-Spin theorem） and Lefschetz fixed points formula and so on,we discuss finite group actions on 4-manifolds. The main research work consists of the following:1. Smooth finite group actions on 4-manifolds;2. Locally linear pseudofree group actions on elliptic surfaces.In Chapter 1, we give a review about the Seiberg-Witten theory and its applications,meanwhile we also introduce the main achievements in this research field obtained by somemathematicians.In Chapter 2, we give some preparations with an emphasis on the basic theory of Seiberg-Witten invariants, we also introduce the technique of the finite dimensional approximation ofSeiberg-Witten theory, basic knowledge of finite group actions on manifolds and K-theory.In Chapter 3, by using the technique of finite dimensional approximation of Seiberg-Wittentheory, the equivariant K-theory and some other methods, we study the problem of topologicalrestriction when there are smooth groups acting on Spin 4-manifolds such as the alternatinggroup A₅, cyclic group Z₆ and symmetric group S₃ with actions of odd type, consequently weimprove Furuta’s 10/8 theorem under the condition of group actions. In particular, when thereis an alternating group A₅ action on a Spin 4-manifold X, we obtain: If X is smooth Spin4-manifold with non-positive signature and b₁（X）=0, denote k=-σ（X）/16 and m=b₂⁺（X）,then 2k+3≤m if b₂⁺（X/<s>）+b₂⁺（X/<t>）≠2b₂⁺（Z/A₅） and b₂⁺（X/<s>）≠0,where <s> and <t> are subgroups of A₅ generated by the elements s=（abcde）∈A₅and t=（abc）∈A₅ respectively. Besides, we get simple expression of the K-theory degree andthe G-index of the equivariant Dirac operator. At last, we discuss a concrete example of thealternating group A₅ action on K3 surfaces.We also study alternating group A₅ actions on the homotopy S²×S² in this chapter. Undersome conditions, we simplify the G-index of the G equivariant Dirac operator to Ind_A5D_X=a（1-2ρ₁+ρ₄）+b（ρ₂-ρ₁）, where a, b are integers. Furthermore, using the same methods, westudy alternating group A₅ actions on the connected sum of n copies of S²×S².In Chapter 4, we apply the G-signature formula, G-Spin theorem and Lefschetz fixed pointsformula to get a totally topological classification of locally linear pseudofree Z₃ actions on elliptic surfaces, we prove that the locally linear pseudofree Z₃ action on elliptic surfaces E（4） belongsto ten types and nine of them can actually be realized by locally linear pseudofree Z₃-actionson elliptic surfaces E（4）, we also give the realization theorem. Meanwhile, by using of the modp vanishing theorem of Seiberg-Witten invariants, we prove the existence of such actions whichcan not be realized as smooth actions on the standard smooth elliptic surfaces.更多还原