Gluck construction
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Contents |
1 Definition
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