Abstract
The problem of embedding spheres in rational surfaces CP 2#n\(\overline {CP} \) 2 is studied. For homology classes u = (b 1 + k, b 2,...,b n) with positive self-intersection numbers, a necessary and sufficient condition to detect its representability is given when k ≤ 5.
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References
Hsing, W., Szczarba, R., On embedding surfaces in four manifolds, Proc. Symp. Pure Math., vol. 22, Providence, Rhade Island: Amer. Math. Soc., 1976, 97–103.
Rohlin, V., Two dimensional submanifolds of four-dimensional manifolds, Founctional Analysis Appl., 1972, 6: 136–138.
Tristram, A. G., Some cobordism invariants for links, Proc. Cambridge Philos. Soc., 1969, 66: 251–264.
Wall, C. T. C., Diffeomorphisms of 4-manifolds, Journal London Math. Soc., 1964, 39: 131–140.
Kervaire, M., Milnor, J., On 2-sphere in 4-manifold, Proc. Nat. Acad. Sci. USA, 1961, 47: 1651–1657.
Fintashal, R., Stern, R., Pseudofree orbifolds, Ann. Math., 1985, 122: 335–364.
Kuga, K., Representing homology class of S 2 x S 2, Topology, 1984, 23: 133–137.
Lawson, T., Representing homology class of almost definite 4-manifolds, Michigan Journal of Math., 1987, 34: 85–91.
Li, B. H., Embedding of surfaces in 4-manifolds, Chinese Science Bulletin, 1991, 36: 2025–2033.
Luo, F., Representing homology classes in CP 2#\(\overline {CP} \) 2, Pacific Journal of Mathematics, 1988, 133: 137–140.
Gan, D., Embedding and immersing of a 2-sphere in indefinite 4-manifolds, Chinese Ann. Math. Ser. B, 1996, 17: 257–262.
Gan, D. Y., Guo, J. H., Embedding and immersing of a 2-sphere in 4-manifolds, Proc. Amer. Math. Soc., 1993, 4: 227–232.
Gao, H. Z., Representing homology classes of almost definite 4-manifolds, Top and Appl., 1993, 52(2): 109–120.
Kikuchi, K., Representing positive homology classes of CP 2#2\(\overline {CP} \) 2 and CP 2#3\(\overline {CP} \) 2, Proc. Amer. Math. Soc., 1993, 117: 861–969.
Kikuchi, K., Positive 2-spheres in 4-manifolds of signature(1, n), Pacific Journal of Math., 1993, 160: 245–258.
Li, B. H., Li, T. J., Minimal genus smooth embedding in S 2 x S 2 and CP 2#n \(\overline {CP} \) 2 with n ≤ 8, Topology, 1998, 37: 575–594.
Li, B. H., Representing nonnegative homology class of CP 2#n \(\overline {CP} \) 2 by minimal genus smooth embedding, Trans. Amer. Math. Soc., 2000, 352(9): 4155–4169.
Lawson, T., Smooth embedding of 2-sphere in 4-manifold, Expo. Math., 1992, 10: 289–309.
Wall, C. T. C., On the orthogonal groups of unimodular quadratic forms, Crelle J., 1963, 213: 122–126.
Li, T. J., Liu, A., General Wall crossing formula, Math Research Letters, 1995, 2: 137–140.
Li, B. H., Li, T. J., Symplectic genus, minimal genus and diffeomorphisms, Asian J. Math., 2002, 6(1): 123–144.
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Gao, H., Guo, R. Homology classes that can be represented by embedded spheres in rational surfaces. Sci. China Ser. A-Math. 47, 431–439 (2004). https://doi.org/10.1360/01ys0348
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DOI: https://doi.org/10.1360/01ys0348