Representing characteristic homology classes of $m\mathbf {C}\mathrm {P}^2 \# n\mathbf {\overline {C}}\mathrm {P}^2$
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- by Jian Han Guo and Dan Yan Gan PDF
- Proc. Amer. Math. Soc. 121 (1994), 1251-1255 Request permission
Abstract:
We prove the following theorems. Theorem 1. If $m,n \geq 1,x \in {H_2}(mC{P^2}\# n{\overline {CP} ^2})$ is a characteristic homology class with ${x^2} = 16l + m - n > 0$ and (1) $m < 3l + 1$ provided $l \geq 0$, or (2) $m < - 19l + 1$ provided $l < 0$. Suppose that the 11/8-conjecture is true. Then x cannot be represented by a smoothly embedded 2-sphere. Theorem 2. Let $m,n \geq 4l > 0,x \in {H_2}(mC{P^2}\# n{\overline {CP} ^2})$ be a primitive characteristic homology class with ${x^2} = \pm 16l + m - n$. Then x can be represented by a smoothly embedded 2-sphere.References
- S. K. Donaldson, An application of gauge theory to four-dimensional topology, J. Differential Geom. 18 (1983), no. 2, 279–315. MR 710056, DOI 10.4310/jdg/1214437665
- S. K. Donaldson, Connections, cohomology and the intersection forms of $4$-manifolds, J. Differential Geom. 24 (1986), no. 3, 275–341. MR 868974, DOI 10.4310/jdg/1214440551
- Michael Freedman and Robion Kirby, A geometric proof of Rochlin’s theorem, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 85–97. MR 520525
- Dan Yan Gan and Jian Han Guo, Smooth embeddings of $2$-spheres in manifolds, J. Math. Res. Exposition 10 (1990), no. 2, 227–232. MR 1057302
- Dan Yan Gan and Jian Han Guo, Embeddings and immersions of a $2$-sphere in $4$-manifolds, Proc. Amer. Math. Soc. 118 (1993), no. 4, 1323–1330. MR 1152976, DOI 10.1090/S0002-9939-1993-1152976-1
- Robert E. Gompf, Infinite families of Casson handles and topological disks, Topology 23 (1984), no. 4, 395–400. MR 780732, DOI 10.1016/0040-9383(84)90002-8
- Michel A. Kervaire and John W. Milnor, On $2$-spheres in $4$-manifolds, Proc. Nat. Acad. Sci. U.S.A. 47 (1961), 1651–1657. MR 133134, DOI 10.1073/pnas.47.10.1651
- Ken’ichi Kuga, Representing homology classes of $S^{2}\times S^{2}$, Topology 23 (1984), no. 2, 133–137. MR 744845, DOI 10.1016/0040-9383(84)90034-X
- Terry Lawson, Representing homology classes of almost definite $4$-manifolds, Michigan Math. J. 34 (1987), no. 1, 85–91. MR 873022, DOI 10.1307/mmj/1029003485
- Bang He Li, Embeddings of surfaces in $4$-manifolds. I, II, Chinese Sci. Bull. 36 (1991), no. 24, 2025–2029, 2030–2033. MR 1150851
- Feng Luo, Representing homology classes of $\textbf {C}\textrm {P}^2\#\;\overline {\textbf {C}\textrm {P}}{}^2$, Pacific J. Math. 133 (1988), no. 1, 137–140. MR 936360, DOI 10.2140/pjm.1988.133.137
- C. T. C. Wall, On the orthogonal groups of unimodular quadratic forms, Math. Ann. 147 (1962), 328–338. MR 138565, DOI 10.1007/BF01440955
- C. T. C. Wall, Diffeomorphisms of $4$-manifolds, J. London Math. Soc. 39 (1964), 131–140. MR 163323, DOI 10.1112/jlms/s1-39.1.131
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 121 (1994), 1251-1255
- MSC: Primary 57R95; Secondary 57R40
- DOI: https://doi.org/10.1090/S0002-9939-1994-1205494-7
- MathSciNet review: 1205494