Representing characteristic homology classes of $m\mathbf {C}\mathrm {P}^2 \# n\mathbf {\overline {C}}\mathrm {P}^2$

Authors:
Jian Han Guo and Dan Yan Gan

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

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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*.

- S. K. Donaldson,
*An application of gauge theory to four-dimensional topology*, J. Differential Geom.**18**(1983), no. 2, 279–315. MR**710056** - S. K. Donaldson,
*Connections, cohomology and the intersection forms of $4$-manifolds*, J. Differential Geom.**24**(1986), no. 3, 275–341. MR**868974** - 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 https://doi.org/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 https://doi.org/10.1016/0040-9383%2884%2990002-8 - 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 https://doi.org/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 https://doi.org/10.1016/0040-9383%2884%2990034-X - Terry Lawson,
*Representing homology classes of almost definite $4$-manifolds*, Michigan Math. J.**34**(1987), no. 1, 85–91. MR**873022**, DOI https://doi.org/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 ${\bf C}{\rm P}^2\#\;\overline {{\bf C}{\rm P}}{}^2$*, Pacific J. Math.**133**(1988), no. 1, 137–140. MR**936360** - C. T. C. Wall,
*On the orthogonal groups of unimodular quadratic forms*, Math. Ann.**147**(1962), 328–338. MR**138565**, DOI https://doi.org/10.1007/BF01440955 - C. T. C. Wall,
*Diffeomorphisms of $4$-manifolds*, J. London Math. Soc.**39**(1964), 131–140. MR**163323**, DOI https://doi.org/10.1112/jlms/s1-39.1.131

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Keywords:
Representing,
characteristic homology class,
primitive

Article copyright:
© Copyright 1994
American Mathematical Society