Gluck construction

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An earlier version of this page was published in the Definitions section of the Bulletin of the Manifold Atlas: screen, print.

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1 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. [Wall1970b] p.232). For any smooth 4-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:
\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 is described as a handlebody, and S is represented by a 2-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 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]).

4 References

5 External references

The Gluck construction on the Wikipedia page about exotic spheres

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