Gluck construction

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1 Definition

Any orientation preserving self diffeomorphism of $S^1\times S^2$ ${{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 $\varphi: S^2\times S^1\to S^2\times S^1$ 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}}. </wikitex> ==Examples== <wikitex>; 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]] </wikitex> ==Some Results== <wikitex>; 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}}). </wikitex> == 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:

$\displaystyle X\mapsto X_{S}= (X- \nu(S))\smile_{\varphi}(S^{2}\times D^{2})$ $\displaystyle X\mapsto X_{S}= (X- \nu(S))\smile_{\varphi}(S^{2}\times D^{2})$

is called the Gluck twisting of $X$ along $S$ . This operation was introduced in [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 $X_{S} \sharp P$ 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]).