Gluck construction

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.

1 Definition

Any orientation preserving self diffeomorphism of $S^1\times S^2$${{Authors:Selman Akbulut}} ==Definition== <wikitex>; Any orientation preserving self diffeomorphism of $S^1\times S^2$ is either isotopic to identity or to the map $\varphi: S^2\times S^1\to S^2\times S^1$ defined by $\varphi(x, y)=(\alpha(y)x,y)$, where $\alpha \in \pi_{1}SO(3)\cong \Z/ 2\Z$ is the generator (e.g. {{cite|W}} p.232). For any smooth $-manifold $X$, and an imbedded 2-sphere in $S\subset X$ with a trivial normal bundle, the operation of removing the regular neighborhood $\nu(S)\cong S^2\times D^2$ of $S$ from $X$ and then regluing it via the nontrivial diffeomorphism: $$ X\mapsto X_{S}= (X- \nu(S))\smile_{\varphi}(S^{2}\times D^{2})$$ is called the ''Gluck twisting of $X$ along $S$''. This operation was introduced in {{cite|Gl}}. </wikitex> ==Examples== <wikitex>; When $X$ is described as a handlebody, and $S$ is represented by a $-handle attached along an unknotted circle with zero framing, then the handlebody of $X_{S}$ is obtained from the handlebody of $X$ by putting one full right (or left) twist to all of the attaching framed circles of the other $-handles going through this circle. <!--\begin{figure}[ht] \begin{center} \includegraphics[width=.48\textwidth]{g} \caption{} <label>fig92</label> \end{center} \end{figure} --> </wikitex> ==Some Results== <wikitex>; It is known that $X_{S} \sharp P$ is diffeomorphic to $X \sharp P$, when $P$ is a copy of $\CP^2$ with either orientation. When $S$ is null-homologous and $X$ is simply connected this operation does not change the homeomorphism type of $X$. It is not known whether a Gluck twisting operation can change the diffeomorphism type of any smooth orientable manifold, while it is known that this is possible in the nonorientable case ({{cite|A1}}). In many instances Gluck twisting of manifolds appear naturally, where this operation do not change their diffeomorphism types (e.g. {{cite|G}}, {{cite|A2}}, {{cite|A3}}, {{cite|AY}}). </wikitex> == References == {{#RefList:}} [[Category:Definitions]]S^1\times S^2$ is either isotopic to identity or to the map $\varphi: S^2\times S^1\to S^2\times S^1$$\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. [W] 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 [Gl].

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 ([A1]). In many instances Gluck twisting of manifolds appear naturally, where this operation do not change their diffeomorphism types (e.g. [G], [A2], [A3], [AY]).

4 References

• [A1] Template:A1
• [A2] Template:A2
• [A3] Template:A3
• [AY] Template:AY
• [G] Template:G
• [Gl] Template:Gl
• [W] Template:W