# 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|Gluck1962}}. |

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

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− | When $X$ is described as a handlebody, and $S$ is represented by a $2$-handle attached along an unknotted circle with zero framing, then the handlebody of $X_{S}$ is obtained from the handlebody of $X$ by putting one full right (or left) twist to all of the attaching framed circles of the other $2$-handles going through this circle. | + | When $X$ is described as a handlebody, and $S$ is represented by a $2$-handle attached along an unknotted circle with zero framing, then the handlebody of $X_{S}$ is obtained from the handlebody of $X$ by putting one full right (or left) twist to all of the attaching framed circles of the other $2$-handles going through this circle. [[Image:Gluck.jpg|thumb|400px|Figure 1]] |

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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 == | ||

{{#RefList:}} | {{#RefList:}} | ||

+ | == External references == | ||

+ | The [[Wikipedia:Gluck_twist#4-dimensional_exotic_spheres_and_Gluck_twists|Gluck construction]] on the Wikipedia page about exotic spheres | ||

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

## Latest revision as of 13:18, 16 May 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 [Gluck1962].

## 2 Examples

## 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 - [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

## 5 External references

The Gluck construction on the Wikipedia page about exotic spheres