Gluck construction

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(Created page with "{{Authors:Selman Akbulut}} ==Definition== <wikitex>; Any orientation preserving self diffeomorphism of $S^1\times S^2$ is either isotopic to identity or to the map $\varphi: ...")
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{{Authors:Selman Akbulut}}
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{{Authors|Selman Akbulut}}
==Definition==
==Definition==
<wikitex>;
<wikitex>;
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|W}} 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|Gl}}.
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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|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|Gl}}.
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==Examples==
==Examples==
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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|A1}}). In many instances Gluck twisting of manifolds appear naturally, where this operation do not change their diffeomorphism types (e.g. {{cite|G}}, {{cite|A2}}, {{cite|A3}}, {{cite|AY}}).
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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|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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</wikitex>
== References ==
== References ==
{{#RefList:}}
{{#RefList:}}
[[Category:Definitions]]
[[Category:Definitions]]

Revision as of 09:44, 15 April 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

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. [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:
\displaystyle  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 [Gl].

2 Examples

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.

3 Some Results

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 ([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

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