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Embeddings and immersions of a $ 2$-sphere in $ 4$-manifolds


Authors: Dan Yan Gan and Jian Han Guo
Journal: Proc. Amer. Math. Soc. 118 (1993), 1323-1330
MSC: Primary 57R40; Secondary 57R42
DOI: https://doi.org/10.1090/S0002-9939-1993-1152976-1
MathSciNet review: 1152976
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Abstract: Let $ M$ be $ C{P^2}\char93 ( - C{P^2})\char93 {P_1}\char93 \cdots \char93 {P_{m + k}}$, where $ {P_1}, \ldots ,{P_{m + k}}$ are copies of $ - C{P^2}$. Let $ h,g,{g_1}, \ldots ,{g_{m + k}}$ be the images of the standard generators of $ {H_2}(C{P^2};Z),\;{H_2}( - C{P^2};Z),\;{H_2}({P_1};Z), \ldots ,{H_2}({P_{m + k}};Z)$ in $ {H_2}(M;Z)$ respectively. Let $ \xi = ph + qg + \sum\nolimits_{i = 1}^m {{r_i}{g_i}} $ be an element of $ {H_2}(M;Z)$. Suppose $ {\xi ^2} = l > 0,\,{p^2} - {q^2} \geqslant 8,\,\vert p\vert - \vert q\vert \geqslant 2$, and $ {r_i} \ne 0$. If $ 2(m + l - 2) \geqslant {p^2} - {q^2}$, then $ \xi $ cannot be represented by a smoothly embedded $ 2$-sphere. If $ 2(m + r + [(l - r - 1)/4] - 1) \geqslant {p^2} - {q^2}$ for some $ r$ with $ 0 \leqslant r \leqslant l - 1$, then for a normal immersion $ f$ of a $ 2$-sphere representing $ \xi $ the number of points of positive self-intersection $ {d_f} \geqslant [(l - r - 1)/4] + 1$.


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  • [B] J. Boardman, Some embeddings of $ 2$-spheres in $ 4$-manifolds, Proc. Cambridge Philos. Soc. 60 (1964), 354-356. MR 0160241 (28:3455)
  • [D] S. Donaldson, An application of gauge theory to four-dimensional topology, J. Differential Geom. 18 (1983), 279-315. MR 710056 (85c:57015)
  • [GG] D. Y. Gan and J. H. Guo, Smooth embeddings of $ 2$-spheres in manifolds, J. Math. Res. Exposition 10 (1990), 227-232. MR 1057302 (92a:57033)
  • [G] R. E. Gompf, Infinite families of Casson handles and topological disks, Topology 23 (1984), 395-400. MR 780732 (86i:57040)
  • [HS] W. C. Hsiang and R. Szczarba, On embedding surfaces in $ 4$-manifolds, Proc. Sympos. Pure Math., vol. 22, Amer. Math. Soc., Providence, RI, 1970, pp. 97-103. MR 0339239 (49:4000)
  • [KM] M. Kervaire and J. Milnor, On $ 2$-spheres in $ 4$-manifolds, Proc. Nat. Acad. Sci. U.S.A. 47 (1961), 1651-1657. MR 0133134 (24:A2968)
  • [K] K. Kuga, Representing homology classes of $ {S^2} \times {S^2}$, Topology 23 (1984), 133-137. MR 744845 (85m:57011)
  • [La] T. Lawson, Representing homology classes of almost definite $ 4$-manifolds, Michigan Math. J. 34 (1987), 85-91. MR 873022 (88d:57024)
  • [Lu] F. Luo, Representing homology classes in $ C{P^2}\char93 \overline {C{P^2}} $, Pacific J. Math. 133 (1988), 137-140. MR 936360 (89h:57031)
  • [R$ _{1}$] V. Rohlin, New results in the theory of $ 4$-dimensional manifolds, Dokl. Akad. Nauk SSSR 84 (1952), 221-224. MR 0052101 (14:573b)
  • [R$ _{2}$] -, Two dimensional submanifolds of four dimensional manifolds, J. Funct. Anal. Appl. 5 (1971), 39-48. MR 0298684 (45:7733)
  • [S] A. Suciu, Immersed spheres in $ C{P^2}$ and $ {S^2} \times {S^2}$, Math. Z. 196 (1987), 51-57. MR 907407 (88j:57038)
  • [T] A. G. Tristram, Some cobordism invariants for links, Proc. Cambridge Philos. Soc. 66 (1969), 251-264. MR 0248854 (40:2104)
  • [W] C. T. C. Wall, Diffeomorphisms of $ 4$-manifolds, J. London Math. Soc. 39 (1964), 131-140. MR 0163323 (29:626)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1993-1152976-1
Keywords: Representing, normal immersion, positive self-intersection
Article copyright: © Copyright 1993 American Mathematical Society

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