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})$$ is called the ''Gluck twisting of $X$ along $S$''. This operation was introduced in {{cite|Gluck1962}}. </wikitex> ==Examples== <wikitex>; When $X$ is described as a handlebody, and $S$ is represented by a $-handle attached along an unknotted circle with zero framing, then the handlebody of $X_{S}$ is obtained from the handlebody of $X$ by putting one full right (or left) twist to all of the attaching framed circles of the other $-handles going through this circle. <!--\begin{figure}[ht] \begin{center} \includegraphics[width=.48\textwidth]{g} \caption{} <label>fig92</label> \end{center} \end{figure} --> </wikitex> ==Some Results== <wikitex>; It is known that $X_{S} \sharp P$ is diffeomorphic to $X \sharp P$, when $P$ is a copy of $\CP^2$ with either orientation. When $S$ is null-homologous and $X$ is simply connected this operation does not change the homeomorphism type of $X$. It is not known whether a Gluck twisting operation can change the diffeomorphism type of any smooth orientable manifold, while it is known that this is possible in the nonorientable case ({{cite|Akbulut1988}}). In many instances Gluck twisting of manifolds appear naturally, where this operation do not change their diffeomorphism types (e.g. {{cite|Gluck1962}}, {{cite|Akbulut1999}}, {{cite|Akbulut2010}}, {{cite|Akbulut&Yasui2012}}). </wikitex> == References == {{#RefList:}} [[Category:Definitions]]S^1\times S^2$ is either isotopic to identity or to the map $\varphi: S^2\times S^1\to S^2\times S^1$$\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$$X$ along $S$$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.

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]).

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 $4$$4$-manifold by Gluck construction to a standard $4$$4$-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