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| EMBEDDED 2-SPHERES IN INDEFINITE 4-MANIFOLDS |
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Citation： |
Gan Danyan.EMBEDDED 2-SPHERES IN INDEFINITE 4-MANIFOLDS[J].Chinese Annals of Mathematics B,1996,17(3):257~262 |
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Net amount： 639 |
Authors： |
Gan Danyan; |
Foundation： |
Project supported by the National Natural Science Foundation of China and by the Zhejiang Natural Science Foundation |
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| Abstract： |
The main results are as follows.
Let $\xi =p\gamma+q_1\delta_1+\cdots +q_n\delta_n$
be an element of $H_2(\cc1)$, where $\gamma,\delta_1,\ldots,
\delta_n$ are standard generators. Suppose $2\leq n\leq 9$.
(1) If $|p|,|q_1|,\cdots,|q_n|\leq 2$ or $|p|=|q_i|$ for some $i$ and
$|q_j|\leq 2$ for $j\neq i$ or $||p|-|q_i||=1$ for some $i$, then
$\xi$ can be represented by a smoothly embedded 2-sphere.
(2) If $\xi$ is a characteristic homology class, then $\xi$ cannot be
represented by a smoothly embedded 2-sphere except for
$\xi^2=16l+1-n$, $l=0$ and $2\leq n\leq 9$; $l=-1$ and $4\leq n\leq 9$;
$l=-2$ and $7\leq n\leq 9$.
(3) If $\xi=d\eta$ for some $d\in\Bbb Z$, where $\eta$ is a primitive
ordinary homology class with $\eta^2=0$, then $\xi$ is eqivalent to
$(d;d,0,\ldots,0)$ and can be represented by a smoothly embedded
2-sphere. |
Keywords： |
Representing, Characteristic, Ordinary, Primitive |
Classification： |
57R95, 57N13, 57R40 |
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