四维流形上的有限群作用
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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A_5、循环群Z_6和奇型三阶对称群S_3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A_5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，b_1(X)=0，令k=-σ(X)／16，m=b_2~+(X)，如果X上具有Spin交错群A_5作用，且满足b_2~+(X／)+b_2~+(X／)≠2b_2~+(X／A_5)，b_2~+(X／)≠0，则2k+3≤m，其中和分别为s=(abcde)∈A_5和t=(abc)∈A_5生成的A_5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A_5在K3曲面上作用的具体例子。
本章还研究了同伦等价于S~2×S~2的Spin 4-流形上的交错群A_5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_(A_5)D_X=a(1-2_(ρ_1+ρ_4))+b(ρ_2-ρ_1)，其中a，b为整数。此外，运用类似的方法我们还研究了n个S~2×S~2的连通和上的交错群A_5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Z_3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Z_3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Z_3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Z_3作用的存在性。
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 A_5, cyclic group Z_6 and symmetric group S_3 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 A_5 action on a Spin 4-manifold X, we obtain: If X is smooth Spin4-manifold with non-positive signature and b_1(X)=0, denote k=-σ(X)/16 and m=b_2~+(X),then 2k+3≤m if b_2~+(X/)+b_2~+(X/)≠2b_2~+(Z/A_5) and b_2~+(X/)≠0,where and are subgroups of A_5 generated by the elements s=(abcde)∈A_5and t=(abc)∈A_5 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 A_5 action on K3 surfaces.
We also study alternating group A_5 actions on the homotopy S~2×S~2 in this chapter. Undersome conditions, we simplify the G-index of the G equivariant Dirac operator to Ind_(A_5)D_X=a(1-2ρ_1+ρ_4)+b(ρ_2-ρ_1), where a, b are integers. Furthermore, using the same methods, westudy alternating group A_5 actions on the connected sum of n copies of S~2×S~2.
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 Z_3 actions on elliptic surfaces, we prove that the locally linear pseudofree Z_3 action on elliptic surfaces E(4) belongsto ten types and nine of them can actually be realized by locally linear pseudofree Z_3-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.
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A_5、循环群Z_6和奇型三阶对称群S_3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A_5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，b_1(X)=0，令k=-σ(X)／16，m=b_2~+(X)，如果X上具有Spin交错群A_5作用，且满足b_2~+(X／
本章还研究了同伦等价于S~2×S~2的Spin 4-流形上的交错群A_5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_(A_5)D_X=a(1-2_(ρ_1+ρ_4))+b(ρ_2-ρ_1)，其中a，b为整数。此外，运用类似的方法我们还研究了n个S~2×S~2的连通和上的交错群A_5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Z_3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Z_3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Z_3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Z_3作用的存在性。
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 A_5, cyclic group Z_6 and symmetric group S_3 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 A_5 action on a Spin 4-manifold X, we obtain: If X is smooth Spin4-manifold with non-positive signature and b_1(X)=0, denote k=-σ(X)/16 and m=b_2~+(X),then 2k+3≤m if b_2~+(X/
We also study alternating group A_5 actions on the homotopy S~2×S~2 in this chapter. Undersome conditions, we simplify the G-index of the G equivariant Dirac operator to Ind_(A_5)D_X=a(1-2ρ_1+ρ_4)+b(ρ_2-ρ_1), where a, b are integers. Furthermore, using the same methods, westudy alternating group A_5 actions on the connected sum of n copies of S~2×S~2.
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 Z_3 actions on elliptic surfaces, we prove that the locally linear pseudofree Z_3 action on elliptic surfaces E(4) belongsto ten types and nine of them can actually be realized by locally linear pseudofree Z_3-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.
引文
[1] S. Akublut, Lectures on Seiberg-Witten invariants. Proceedings of the Gokova Geometry Topology conference (1996), 95-118.
[2] C. Allday and V. Puppe, Cohomological Methods in Transformations Groups, Cambidge studies in Advanced Mathematics 32, Cambridge Univ. Press(1993).
[3] M. F. Atiyah, Bott periodicity and the index of elliptic operators. Quart. J. Math. Oxford, 19(2), 1968.
[4] M. F. Atiyah and F. Hirzebruch, Spin-manifolds and group actions, Essays on topology and related topics, Memoires dedie a George de Rham (ed. A. Haefliger and R. Narashimhan), Springer-Verlag (1970), 18-28.
[5] M. F. Atiyah and R. Bott. A Lefschetz fixed point formula for elliptic complexes II Applications, Ann. Math. 88(1968), 451-491.
[6] S. Baldridge, Seiberg-Witten vanishing theorem for S~1-manifolds with fixed points. Pacific J. Math. 217 (2004), no. 1, 1-10.
[7] S. Baldridge, Seiberg-Witten invariants, orbifolds, and circle actions. Trans. Amer. Math. Soc. 355 (2003), no. 4, 1669-1697.
[8] S. Baldridge, Seiberg-Witten invariants of 4-manifolds with free circle actions. Commun. Contemp. Math. 3 (2001), no. 3, 341-353.
[9] S. Bauer and M. Furuta, A stable cohomotopy refinement of Seiberg-Witten invariants I, Invent. Math. 155(2004), 1-19.
[10] S. Bauer, A stable cohomotopy refinement of Seiberg-Witten invariants II, Invent. Math. 155(2004), 21-40.
[11] C. Bohr, On the signatures of even 4-manifolds, Math. Proc. Cambridge Philos. Soc. 132(2002), no. 3, 453-469.
[12] J. Bryan, Seiberg-Witten theory and Z2P-actions on spin 4-manifolds, Math. Res. Letters. 5(1998), 165-183.
[13] Y. S. Cho and Y. H. Hong, Seiberg-Witten invariants and (anti-)symplectic involutions. Glasg. Math. J. 45 (2003), no. 3, 401-413.
[14] Daniel Ruberman, Involutions on Spin 4-manifolds. Proceedings of the American Mathematical Society. 123 (1995), no. 2, 593-596.
[15] T. torn Dieck, Transformation Groups and Representation Theory. Nmber 766 in Lecture Notes in Mathematics. Springer-Verlag, 1970.
[16] S. K. Donaldson, Connections, cohomology and the intersection forms of four manifolds. J. Differential Geom. 24 (1986), 275-341.
[17] S. K. Donaldson, The orientation of Yang-Mills moduli spaces and four manifold topology. J. Differential Geom. 26 (1987), 397-428.
[18] S. K. Donaldson, An application of gauge theory to four dimensional topology. Journal of Differential Geometry 18 (1983), 279-315.
[19] A. L. Edmonds, Aapects of group actions on four-manifolds, Topology Appl. 31(1989) no. 2, 109- 124.
[20] A. L. Edmonds and J. H. Ewing, Realizing forms and fixed point data in dimension four, Amer. J. Math. 114(1992), 1103-1126.
[21] F. Fang, Smooth group actions on 4-manifolds and Seiberg-Witten invariants: II, Int. J. Math. 9(1998), 957-973.
[22] F. Fang, Smooth group actions on 4-manifolds and Seiberg-Witten theory, Diff. Geom. and its Applications, 14(2001), 1-14.
[23] R. Fintshel and R. Stern, Rational blowdown of smooth 4-manifolds, J. Diff. Geom. 46(1997), 181-235.
[24] R. Fintshel and R. Stern, Definite 4-manifolds. Journal of Differential Geometry 28 (1988), 133-141.
[25] M. H. Freedman, The topology of four-dimensional manifolds, J. Diff. Geom. 17(1982), 357-453.
[26] R. Friedman and J. Morgan, Algebraic surfaces and Seiberg-Witten invariants, J. Algebraic Geom. 6(1997), 445-479.
[27] R. Friedman and J. Morgan, Obstruction bundles, semi-regularity and Seiberg-Witten invariants, Comm. Anal. Geom. 7(1999), 451-495.
[28] M. Furuta, Monopole equation and 11/8 conjecture, Math. Res. Letter, 8(2001),157-176.
[29] M. Furuta, Finite dimensional approximations in geometry. Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), 395-403, Higher Ed. Press, Beijing, 2002.
[30] M. Furuta, Y. Kametani, The Seiberg-Witten equations and equivariant e-invariants, priprint, 2001.
[31] M. Furuta and Y. Kametani, Equivariant maps between sphere bundles over tori and KO*-degree, preprint.
[32] R. E. Gompf and A. I. Stipsicz, 4-manifolds and Kirby calculus, American Mathematical Society, Providence, RI, 1999.
[33] R. E. Gompf, Nuclei of elliptic surfaces, Topology 30(1991), 479-511.
[34] J. H. Kim, On spin Z2P-actions on spin 4-manifolds, Topology and its Applications. 108(2000), 197-215.
[35] K. Kiyono and Ximin Liu, On spin alternating group actions on spin 4-manifolds, J. of Korean Math. Soc. 43(2006), no. 6, 1183-1197.
[36] J. Morgan and Z. Szabo, Homotopy K3 surfaces and mod 2 Seiberg-Witten invariants, Math. Research Letters, 4(1997), 17-21.
[37] P. B. Kronheimer and T. S. Mrowka, The genus of imbedded surfaces in the projective plane, Math. Res. Letters 1(1994), 797-808.
[38] H. B. Lawson, Jr. and M.-L. Michelsohn, Spin geometry, Princeton Mathematical Series, Volume 38, Princeton University Press, Princeton, NJ, 1989.
[39] R. Lee and T. J. Li, Intersection forms of non-spin four manifolds, Math. Ann. 319(2001), 311-318.
[40] Ximin Liu, On S_3-actions on Spin 4-manifolds, Carpathian J. Math. 21(2005), 137-142.
[41] Ximin Liu and N. Nakamura, Pseudofree Z_3actions on K3 surfaces, Proc, of Amer. Math. Soc. 135 (2007), no. 3, 903-910
[42] Y. Matsumoto, On the bounding genus of homology 3-spheres, J. Fac. Sci. Univ. Tokyo Sect. IA. Math. 29(1982), 287-318.
[43] N. Minami, The G-join theorem - an unbased G-Freudenthal theorem, preprint.
[44] B. Moishezon, Complex surfaces and connected sums of complex projective planes, Lecture Notes in Math. 603, Springer, (1977).
[45] John D. Moore, Lectures on Seiberg-Witten Invariants, Springer-Verlag. 1996.
[46] J. W. Morgan, The Seiberg-Witten equations and applications to the topology of smooth four-manifolds, Math. Notes 44, Princeton University Press,1996.
[47] J. W. Morgan and Zoltan Szabo, Homotopy K3 surfaces and mod 2 Seiberg-Witten invariants, Math. Res. Lett. 4(1997) 17-21.
[48] N. Nakamura, Mod p vanishing theorem of Seiberg-Witten invariants for 4-manifolds with Z/p-actions, Asian J. Math., to appear.
[49] N. Nakamura, A free Z_p-action and the Seiberg-Witten invariants, J. Korean Math. Soc. 39(2002), no. 1, 103-117.
[50] Y. B. Ruan and S. G. Wang, Seiberg-Witten invariants and double covers of 4-manifolds. Comm. Anal. Geom. 8(2000), no. 3, 477-515.
[51] V. A. Rohlin, New results in the theory of four dimensional manifolds. Dok. Akad. Nauk. USSR 84 (1952), 221-224.
[52] N. Seiberg and E. Witten, electric-magnetic duality, monopole condensation, and confinement in N = 2 supersymmetric Yang-Mills theory. Nuclear Physics B, 1994, B426(1):19-52.
[53] J. P. Serre, Linear Representation of Finite Groups, Springer-Verlag, New York, 1977.
[54] P. Shanahan, The Atiyah-Singer index theorem, Lecture Notes in Mathematics, volume 638, Sp-inger, Berlin, 1978.
[55] S. Stolz, The level of real projective spaces, Comment. Math. Helvetici, 64(1989), 661-674.
[56] M. Szymic, Bauer-Furuta invariant and Galois symmetries, preprint.
[57] C. H. Tabes, The Seiberg-Witten invariants and symplectic forms, Math. Res. Letters 1(1994), 809-822.
[58] C. H. Tabes, The Seiberg-Witten and Gromov invariants, Math. Res. Letters 2(1995), 221-238.
[59] H. Tokui, Msc thesis, University of Tokyo, January 1997.
[60] M. Ue, On the topology of elliptic surfaces-a survey, Amer. Math. Soc. Transl. 160(1994), 95-123.
[61] E. Witten, Monopoles and four-manifolds, Math. Research Letters 1(1994), 769-796.
[2] C. Allday and V. Puppe, Cohomological Methods in Transformations Groups, Cambidge studies in Advanced Mathematics 32, Cambridge Univ. Press(1993).
[3] M. F. Atiyah, Bott periodicity and the index of elliptic operators. Quart. J. Math. Oxford, 19(2), 1968.
[4] M. F. Atiyah and F. Hirzebruch, Spin-manifolds and group actions, Essays on topology and related topics, Memoires dedie a George de Rham (ed. A. Haefliger and R. Narashimhan), Springer-Verlag (1970), 18-28.
[5] M. F. Atiyah and R. Bott. A Lefschetz fixed point formula for elliptic complexes II Applications, Ann. Math. 88(1968), 451-491.
[6] S. Baldridge, Seiberg-Witten vanishing theorem for S~1-manifolds with fixed points. Pacific J. Math. 217 (2004), no. 1, 1-10.
[7] S. Baldridge, Seiberg-Witten invariants, orbifolds, and circle actions. Trans. Amer. Math. Soc. 355 (2003), no. 4, 1669-1697.
[8] S. Baldridge, Seiberg-Witten invariants of 4-manifolds with free circle actions. Commun. Contemp. Math. 3 (2001), no. 3, 341-353.
[9] S. Bauer and M. Furuta, A stable cohomotopy refinement of Seiberg-Witten invariants I, Invent. Math. 155(2004), 1-19.
[10] S. Bauer, A stable cohomotopy refinement of Seiberg-Witten invariants II, Invent. Math. 155(2004), 21-40.
[11] C. Bohr, On the signatures of even 4-manifolds, Math. Proc. Cambridge Philos. Soc. 132(2002), no. 3, 453-469.
[12] J. Bryan, Seiberg-Witten theory and Z2P-actions on spin 4-manifolds, Math. Res. Letters. 5(1998), 165-183.
[13] Y. S. Cho and Y. H. Hong, Seiberg-Witten invariants and (anti-)symplectic involutions. Glasg. Math. J. 45 (2003), no. 3, 401-413.
[14] Daniel Ruberman, Involutions on Spin 4-manifolds. Proceedings of the American Mathematical Society. 123 (1995), no. 2, 593-596.
[15] T. torn Dieck, Transformation Groups and Representation Theory. Nmber 766 in Lecture Notes in Mathematics. Springer-Verlag, 1970.
[16] S. K. Donaldson, Connections, cohomology and the intersection forms of four manifolds. J. Differential Geom. 24 (1986), 275-341.
[17] S. K. Donaldson, The orientation of Yang-Mills moduli spaces and four manifold topology. J. Differential Geom. 26 (1987), 397-428.
[18] S. K. Donaldson, An application of gauge theory to four dimensional topology. Journal of Differential Geometry 18 (1983), 279-315.
[19] A. L. Edmonds, Aapects of group actions on four-manifolds, Topology Appl. 31(1989) no. 2, 109- 124.
[20] A. L. Edmonds and J. H. Ewing, Realizing forms and fixed point data in dimension four, Amer. J. Math. 114(1992), 1103-1126.
[21] F. Fang, Smooth group actions on 4-manifolds and Seiberg-Witten invariants: II, Int. J. Math. 9(1998), 957-973.
[22] F. Fang, Smooth group actions on 4-manifolds and Seiberg-Witten theory, Diff. Geom. and its Applications, 14(2001), 1-14.
[23] R. Fintshel and R. Stern, Rational blowdown of smooth 4-manifolds, J. Diff. Geom. 46(1997), 181-235.
[24] R. Fintshel and R. Stern, Definite 4-manifolds. Journal of Differential Geometry 28 (1988), 133-141.
[25] M. H. Freedman, The topology of four-dimensional manifolds, J. Diff. Geom. 17(1982), 357-453.
[26] R. Friedman and J. Morgan, Algebraic surfaces and Seiberg-Witten invariants, J. Algebraic Geom. 6(1997), 445-479.
[27] R. Friedman and J. Morgan, Obstruction bundles, semi-regularity and Seiberg-Witten invariants, Comm. Anal. Geom. 7(1999), 451-495.
[28] M. Furuta, Monopole equation and 11/8 conjecture, Math. Res. Letter, 8(2001),157-176.
[29] M. Furuta, Finite dimensional approximations in geometry. Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), 395-403, Higher Ed. Press, Beijing, 2002.
[30] M. Furuta, Y. Kametani, The Seiberg-Witten equations and equivariant e-invariants, priprint, 2001.
[31] M. Furuta and Y. Kametani, Equivariant maps between sphere bundles over tori and KO*-degree, preprint.
[32] R. E. Gompf and A. I. Stipsicz, 4-manifolds and Kirby calculus, American Mathematical Society, Providence, RI, 1999.
[33] R. E. Gompf, Nuclei of elliptic surfaces, Topology 30(1991), 479-511.
[34] J. H. Kim, On spin Z2P-actions on spin 4-manifolds, Topology and its Applications. 108(2000), 197-215.
[35] K. Kiyono and Ximin Liu, On spin alternating group actions on spin 4-manifolds, J. of Korean Math. Soc. 43(2006), no. 6, 1183-1197.
[36] J. Morgan and Z. Szabo, Homotopy K3 surfaces and mod 2 Seiberg-Witten invariants, Math. Research Letters, 4(1997), 17-21.
[37] P. B. Kronheimer and T. S. Mrowka, The genus of imbedded surfaces in the projective plane, Math. Res. Letters 1(1994), 797-808.
[38] H. B. Lawson, Jr. and M.-L. Michelsohn, Spin geometry, Princeton Mathematical Series, Volume 38, Princeton University Press, Princeton, NJ, 1989.
[39] R. Lee and T. J. Li, Intersection forms of non-spin four manifolds, Math. Ann. 319(2001), 311-318.
[40] Ximin Liu, On S_3-actions on Spin 4-manifolds, Carpathian J. Math. 21(2005), 137-142.
[41] Ximin Liu and N. Nakamura, Pseudofree Z_3actions on K3 surfaces, Proc, of Amer. Math. Soc. 135 (2007), no. 3, 903-910
[42] Y. Matsumoto, On the bounding genus of homology 3-spheres, J. Fac. Sci. Univ. Tokyo Sect. IA. Math. 29(1982), 287-318.
[43] N. Minami, The G-join theorem - an unbased G-Freudenthal theorem, preprint.
[44] B. Moishezon, Complex surfaces and connected sums of complex projective planes, Lecture Notes in Math. 603, Springer, (1977).
[45] John D. Moore, Lectures on Seiberg-Witten Invariants, Springer-Verlag. 1996.
[46] J. W. Morgan, The Seiberg-Witten equations and applications to the topology of smooth four-manifolds, Math. Notes 44, Princeton University Press,1996.
[47] J. W. Morgan and Zoltan Szabo, Homotopy K3 surfaces and mod 2 Seiberg-Witten invariants, Math. Res. Lett. 4(1997) 17-21.
[48] N. Nakamura, Mod p vanishing theorem of Seiberg-Witten invariants for 4-manifolds with Z/p-actions, Asian J. Math., to appear.
[49] N. Nakamura, A free Z_p-action and the Seiberg-Witten invariants, J. Korean Math. Soc. 39(2002), no. 1, 103-117.
[50] Y. B. Ruan and S. G. Wang, Seiberg-Witten invariants and double covers of 4-manifolds. Comm. Anal. Geom. 8(2000), no. 3, 477-515.
[51] V. A. Rohlin, New results in the theory of four dimensional manifolds. Dok. Akad. Nauk. USSR 84 (1952), 221-224.
[52] N. Seiberg and E. Witten, electric-magnetic duality, monopole condensation, and confinement in N = 2 supersymmetric Yang-Mills theory. Nuclear Physics B, 1994, B426(1):19-52.
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