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Let $X$ be a 4-manifold and $\Sigma_g\subset X$ be an embedded closed orientable genus = $g$ surface. Suppose $\Sigma_g\subset X$ has a trivial closed normal bundle $N(\Sigma_g) = \Sigma_g\times D^2$. For a given self-diffeomorphism $h$ on $\Sigma_g\times S^1$, we can define a new 4-manifold:

$$X(\Sigma_g,h):=(X-\text{Int}N(\Sigma_g))\cup_{h}(\Sigma_g\times D^2)$$

For example, in the case of $g=0$ this operation is called Gluck construction, and in the case of $g=1$ it is well known as (generalized) logarithmic transformation. My question is:

Is there any research in the case of $g\geq2$?

Thank you for your help.

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For $g\geq 2$, this construction doesn't yield anything new. In this situation, any self-diffeomorphism of $\Sigma_g \times S^1$ extends to a self-diffeomorphism of $\Sigma_g \times D^2$, and hence your $X$ is unchanged.

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  • $\begingroup$ @Ruberman Thank you for your helpful comment. May I ask you that why any self-diffeomorphism of $\Sigma_g\times S^1$ extends to a self diffeomorphism of $\Sigma_g\times D^2$ for $g\geq2$ ? $\endgroup$
    – Shinichiro Nakamura
    Commented May 20, 2017 at 6:51
  • $\begingroup$ @ShinichiroNakamura You can probably find this in various books on 3-manifolds, eg Hempel. One proof can be based on the assertion that any self-homeomorphism is fiber-preserving (ie takes each circle to a circle). A reference is Scott's paper, The geometries of 3-manifolds, theorem 3.9. He refers to older work of Waldhausen and Orlik-Vogt-Zieschang as the original sources. $\endgroup$
    – Danny Ruberman
    Commented May 20, 2017 at 15:46
  • $\begingroup$ @Ruberman Thank you, I understood. Well, it seems that this question was elementary level... I will study harder and harder. $\endgroup$
    – Shinichiro Nakamura
    Commented May 21, 2017 at 5:28

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