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

Gan, Dan Yan; Guo, Jian Han

doi:10.1090/S0002-9939-1993-1152976-1

Embeddings and immersions of a $2$-sphere in $4$-manifolds
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by Dan Yan Gan and Jian Han Guo PDF

Proc. Amer. Math. Soc. 118 (1993), 1323-1330 Request permission

Abstract:

Let $M$ be $C{P^2}\# ( - C{P^2})\# {P_1}\# \cdots \# {P_{m + k}}$, 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 ${\xi ^2} = l > 0, {p^2} - {q^2} \geqslant 8, |p| - |q| \geqslant 2$, 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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