# Gluck construction

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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