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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Representing nonnegative homology classes
of ${\mathbb C}P^2\#n{\overline{{\mathbb C}P}}{}^2$ by minimal genus smooth embeddings

Author: Bang-He Li
Journal: Trans. Amer. Math. Soc. 352 (2000), 4155-4169
MSC (1991): Primary 57R95, 57R40
Published electronically: May 21, 1999
MathSciNet review: 1637082
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Abstract: For any nonnegative class $\xi$ in $H_2({\mathbb C}P^2\#n{\overline{{\mathbb C}P}}{}^2, {\mathbf Z})$, the minimal genus of smoothly embedded surfaces which represent $\xi$ is given for $n\leq 9$, and in some cases with $n\geq 10$, the minimal genus is also given. For the finiteness of orbits under diffeomorphisms with minimal genus $g$, we prove that it is true for $n\leq 8$ with $g\geq 1$ and for $n\leq 9$ with $g\geq 2$.

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  • [D1] S. Donaldson, The orientation of Yang-Mills moduli spaces and $4$-manifold topology, J. Differential Geom. 26 (1987), 397-428. MR 88j:57020
  • [D2] -, The Seiberg-Witten equations and $4$-manifold topology, Bull. (N.S.) Amer. Math. Soc. 33 (1996), 45-70. MR 96k:57033
  • [FM] R. Friedman and J. W. Morgan, On the diffeomorphism types of certain algebraic surfaces, I, J. Differential Geom. 27 (1988), 297-369. MR 89d:57046
  • [FS] R. Fintushel and R. J. Stern, Immersed spheres in 4-manifolds and the immersed Thom conjecture, Turkish J. Math. 19 (1995), 145-157. MR 96j:57036
  • [G1] H. Z. Gao, Representing homology classes of 4-manifolds, Topology and App. 52(2) (1993), 109-120. MR 94j:57020
  • [G2] D. Y. Gan, Embedded 2-spheres in indefinite 4-manifolds, Chinese Ann. Math. Ser. B 17 (1996), 257-262. MR 97j:57056
  • [HS] W. C. Hsiang and R. Szczarba, On embedding surfaces in four-manifold, Proc. Symp. Pure Math., vol. 22, Amer. Math. Soc., Providence, RI, 1970, pp. 97-103. MR 49:4000
  • [K1] K. Kikuchi, Representing positive homology classes of ${\mathbb C}P^2\# 2\overline{{\mathbb C}P}^2$ and ${\mathbb C}P^2\# 3\overline{{\mathbb C}P}^2$, Proc. Amer. Math. Soc. 117 (1993), 861-969. MR 93d:57065
  • [K2] -, Positive $2$-spheres in $4$-manifolds of signature $(1, n)$, Pacific J. Math. 160 (1993), 245-258. MR 94f:57027
  • [KM1] M. Kervaire and W. Milnor, On $2$-spheres in $4$-manifolds, Proc. Nat. Acad. Sci. USA 47 (1961), 1651-1657. MR 24:A2968
  • [KM2] P. Kronheimer and T. Mrowka, The genus of embedded surfaces in the projective plane, Math. Research Letters 1 (1994), 797-808. MR 96a:57073
  • [L] T. Lawson, Smooth embeddings of $2$-spheres in $4$-manifolds, Expo. Math. 10 (1992), 289-309. MR 93m:57041
  • [Li] B. H. Li, Embeddings of surfaces in $4$-manifolds, Chinese Science Bulletin 36 (1991), 2025-2033. MR 93d:57037
  • [LiL1] B. H. Li and T. J. Li, Minimal genus smooth embeddings in $S^2\times S^2$ and $CP^2\# n\overline{CP}^2$ with $n\leq 8$, Topology 37 (1998), 575-594. MR 99b:57059
  • [LiL2] -, Minimal genus embeddings of surfaces in $S^2$-bundles over surfaces, Math. Research Letters 4 (1997), 379-384. MR 98h:57062
  • [LL1] T. J. Li and A. Liu, General Wall crossing formula, Math. Research Letters 2 (1995), 797-810. MR 96m:57053
  • [LL2] -, Symplectic structures on ruled surfaces and generalized adjunction formula, Math. Research Letters 2 (1995), 453-471. MR 96m:57052
  • [MST] J. Morgan, Z. Szabo and C. H. Taubes, A product formula for the Seiberg-Witten invariants and the generalized Thom conjecture, J. Differential Geom. 44 (1996), 706-788. MR 97m:57052
  • [Ro] V. Rohlin, Two-dimensional submanifolds of four-dimensional manifolds, Functional Analysis Appl. 6 (1972), 136-138.
  • [Ru] D. Ruberman, The minimal genus of an embedded surface of nonnegative square in a rational surface, Turkish J. Math. 20 (1996), 129-133. MR 97k:57036
  • [T1] C. H. Taubes, The Seiberg-Witten invariants and symplectic forms, Math. Research Letters 1 (1994), 809-822. MR 95j:57039
  • [T2] -, More constraints on symplectic forms from Seiberg-Witten invariants, Math. Research Letters 2 (1995), 9-13. MR 96a:57075
  • [T3] -, The Seiberg-Witten invariants and The Gromov invariants, Math. Research Letters 2 (1995), 221-238. MR 96a:57076
  • [T4] -, Talks given at Harvard.
  • [Ue] M. Ue, Embedded surfaces in blowing up rational and elliptic surfaces, (abstract for `` Art of Low-Dimensional Topology III"), Preprint of the Division of Mathematics, Kyoto Univ.
  • [Wa] C. T. C. Wall, Diffeomorphisms of $4$-manifolds, J. London Math. Soc. 39 (1964), 131-140. MR 27:6278
  • [Wi] E. Witten, Monopoles and Four-Manifolds, Math. Research Letters 1 (1994), 769-796. MR 96d:57035

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Additional Information

Bang-He Li
Affiliation: Institute of Systems Science, Chinese Academy of Sciences, Beijing 100080, Peoples Republic of China

Received by editor(s): March 25, 1998
Published electronically: May 21, 1999
Additional Notes: The author is supported partially by the Tianyuan Foundation of Peoples Republic of China
Article copyright: © Copyright 2000 American Mathematical Society

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