# 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

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