# Gluck construction

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==Definition== | ==Definition== | ||

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− | Any orientation preserving self diffeomorphism of $S^1\times S^2$ is either isotopic to identity or to the map $\varphi: S^2\times S^1\to S^2\times S^1$ defined by $\varphi(x, y)=(\alpha(y)x,y)$, where $\alpha \in \pi_{1}SO(3)\cong \Z/ 2\Z$ is the generator (e.g. {{cite| | + | Any orientation preserving self diffeomorphism of $S^1\times S^2$ is either isotopic to identity or to the map $\varphi: S^2\times S^1\to S^2\times S^1$ defined by $\varphi(x, y)=(\alpha(y)x,y)$, where $\alpha \in \pi_{1}(SO(3))\cong \Z/ 2\Z$ is the generator (e.g. {{cite|Wall1970b}} p.232). For any smooth $4$-manifold $X$, and an imbedded 2-sphere in $S\subset X$ with a trivial normal bundle, the operation of removing the regular neighborhood $\nu(S)\cong S^2\times D^2$ of $S$ from $X$ and then regluing it via the nontrivial diffeomorphism: $$ X\mapsto X_{S}= (X- \nu(S))\smile_{\varphi}(S^{2}\times D^{2})$$ is called the ''Gluck twisting of $X$ along $S$''. This operation was introduced in {{cite|Gl}}. |

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==Examples== | ==Examples== | ||

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==Some Results== | ==Some Results== | ||

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− | It is known that $X_{S} \sharp P$ is diffeomorphic to $X \sharp P$, when $P$ is a copy of $\CP^2$ with either orientation. When $S$ is null-homologous and $X$ is simply connected this operation does not change the homeomorphism type of $X$. It is not known whether a Gluck twisting operation can change the diffeomorphism type of any smooth orientable manifold, while it is known that this is possible in the nonorientable case ({{cite| | + | It is known that $X_{S} \sharp P$ is diffeomorphic to $X \sharp P$, when $P$ is a copy of $\CP^2$ with either orientation. When $S$ is null-homologous and $X$ is simply connected this operation does not change the homeomorphism type of $X$. It is not known whether a Gluck twisting operation can change the diffeomorphism type of any smooth orientable manifold, while it is known that this is possible in the nonorientable case ({{cite|Akbulut1988}}). In many instances Gluck twisting of manifolds appear naturally, where this operation do not change their diffeomorphism types (e.g. {{cite|Gluck1962}}, {{cite|Akbulut1999}}, {{cite|Akbulut2010}}, {{cite|Akbulut&Yasui2012}}). |

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== References == | == References == | ||

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[[Category:Definitions]] | [[Category:Definitions]] |

## Revision as of 10:44, 15 April 2013

An earlier version of this page was published in the Definitions section of the Bulletin of the Manifold Atlas: screen, print. You may view the version used for publication as of 12:18, 16 May 2013 and the changes since publication. |

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## Contents |

## 1 Definition

*Gluck twisting of along*. This operation was introduced in [Gl].

## 2 Examples

When is described as a handlebody, and is represented by a -handle attached along an unknotted circle with zero framing, then the handlebody of is obtained from the handlebody of by putting one full right (or left) twist to all of the attaching framed circles of the other -handles going through this circle.

## 3 Some Results

It is known that is diffeomorphic to , when is a copy of with either orientation. When is null-homologous and is simply connected this operation does not change the homeomorphism type of . It is not known whether a Gluck twisting operation can change the diffeomorphism type of any smooth orientable manifold, while it is known that this is possible in the nonorientable case ([Akbulut1988]). In many instances Gluck twisting of manifolds appear naturally, where this operation do not change their diffeomorphism types (e.g. [Gluck1962], [Akbulut1999], [Akbulut2010], [Akbulut&Yasui2012]).

## 4 References

- [Akbulut&Yasui2012] S. Akbulut and K. Yasui,
*Gluck twisting 4-manifolds with odd intersection form*, (2012). Available at the arXiv:1205.6038. - [Akbulut1988] S. Akbulut,
*Constructing a fake -manifold by Gluck construction to a standard -manifold*, Topology**27**(1988), no.2, 239–243. MR948185 (89j:57014) Zbl 0649.57011 - [Akbulut1999] S. Akbulut,
*Scharlemann's manifold is standard*, Ann. of Math. (2)**149**(1999), no.2, 497–510. MR1689337 (2000d:57033) Zbl 0931.57016 - [Akbulut2010] S. Akbulut,
*Cappell-Shaneson homotopy spheres are standard*, Ann. of Math. (2)**171**(2010), no.3, 2171–2175. MR2680408 (2011i:57024) Zbl 1216.57017 - [Gl] Template:Gl
- [Gluck1962] H. Gluck,
*The embedding of two-spheres in the four-sphere*, Trans. Amer. Math. Soc.**104**(1962), 308–333. MR0146807 (26 #4327) Zbl 0111.18804 - [Wall1970b] C. T. C. Wall,
*Surgery on compact manifolds*, Academic Press, London, 1970. MR0431216 (55 #4217) Zbl 0935.57003