# Gluck construction

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## Contents |

## 1 Definition

*Gluck twisting of along*. This operation was introduced in [Gluck1962].

## 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 - [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