Gluck construction

Revision as of 10:44, 15 April 2013 (view source) Diarmuid Crowley (Talk | contribs) ← Older edit		Latest revision as of 14:18, 16 May 2013 (view source) Diarmuid Crowley (Talk | contribs)
(6 intermediate revisions by one user not shown)
Line 2:		Line 2:
	==Definition==		==Definition==
	<wikitex>;		<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 $4$-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: $$ X\mapsto X_{S}= (X- \nu(S))\smile_{\varphi}(S^{2}\times D^{2})$$ is called the ''Gluck twisting of $X$ along $S$''. This operation was introduced in {{cite|Gl}}.	+	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 $4$-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: $$ X\mapsto X_{S}= (X- \nu(S))\smile_{\varphi}(S^{2}\times D^{2})$$ is called the ''Gluck twisting of $X$ along $S$''. This operation was introduced in {{cite|Gluck1962}}.
	</wikitex>		</wikitex>
		+
	==Examples==		==Examples==
	<wikitex>;		<wikitex>;
−	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.	+	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. [[Image:Gluck.jpg|thumb|400px|Figure 1]]
−	<!--\begin{figure}[ht] \begin{center} \includegraphics[width=.48\textwidth]{g} \caption{} <label>fig92</label>	+
−	\end{center} \end{figure} -->	+
	</wikitex>		</wikitex>
		+
	==Some Results==		==Some Results==
	<wikitex>;		<wikitex>;
Line 16:		Line 16:
	== References ==		== References ==
	{{#RefList:}}		{{#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]]		[[Category:Definitions]]

---

Latest revision as of 14:18, 16 May 2013

The user responsible for this page is Selman Akbulut. No other user may edit this page at present.

Contents 1 Definition 2 Examples 3 Some Results 4 References 5 External references

1 Definition

Any orientation preserving self diffeomorphism of ${{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|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|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 $\varphi: S^2\times S^1\to S^2\times S^1$ 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: 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.

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

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

5 External references

The Gluck construction on the Wikipedia page about exotic spheres