# Gluck construction

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## 1 Definition

Any orientation preserving self diffeomorphism of $S^1\times S^2$${{Authors|Selman Akbulut}} ==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. {{cite|Wall1970b}} p.232). For any smooth -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}}. ==Examples== ; When X is described as a handlebody, and S is represented by a -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 -handles going through this circle. [[Image:Gluck.jpg|thumb|400px|Figure 1]] ==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 ({{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}}). == References == {{#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]]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$$\varphi: S^2\times S^1\to S^2\times S^1$ defined by $\varphi(x, y)=(\alpha(y)x,y)$$\varphi(x, y)=(\alpha(y)x,y)$, where $\alpha \in \pi_{1}(SO(3))\cong \Z/ 2\Z$$\alpha \in \pi_{1}(SO(3))\cong \Z/ 2\Z$ is the generator (e.g. [Wall1970b] p.232). For any smooth $4$$4$-manifold $X$$X$, and an imbedded 2-sphere in $S\subset X$$S\subset X$ with a trivial normal bundle, the operation of removing the regular neighborhood $\nu(S)\cong S^2\times D^2$$\nu(S)\cong S^2\times D^2$ of $S$$S$ from $X$$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$$X$ along $S$$S$. This operation was introduced in [Gluck1962].

## 2 Examples

When $X$$X$ is described as a handlebody, and $S$$S$ is represented by a $2$$2$-handle attached along an unknotted circle with zero framing, then the handlebody of $X_{S}$$X_{S}$ is obtained from the handlebody of $X$$X$ by putting one full right (or left) twist to all of the attaching framed circles of the other $2$$2$-handles going through this circle.
Figure 1

## 3 Some Results

It is known that $X_{S} \sharp P$$X_{S} \sharp P$ is diffeomorphic to $X \sharp P$$X \sharp P$, when $P$$P$ is a copy of $\CP^2$$\CP^2$ with either orientation. When $S$$S$ is null-homologous and $X$$X$ is simply connected this operation does not change the homeomorphism type of $X$$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]).