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
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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