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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
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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DOI: http://dx.doi.org/10.1090/S0002-9939-1993-1152976-1
PII: S 0002-9939(1993)1152976-1
Keywords: Representing, normal immersion, positive self-intersection
Article copyright: © Copyright 1993 American Mathematical Society