Akbulut cork

{{Short description|Structure in topology}}

In topology, an Akbulut cork is a structure that is frequently used to show that in 4 dimensions, the smooth h-cobordism theorem fails. It was named after Turkish mathematician Selman Akbulut.{{cite book|first1=Robert E.|last1= Gompf |author1-link=Robert Gompf| first2=András I. |last2=Stipsicz|title= 4-manifolds and Kirby calculus |page=357|series= Graduate Studies in Mathematics|volume= 20|publisher= American Mathematical Society | location=Providence, RI |year=1999 | isbn=0-8218-0994-6|doi=10.1090/gsm/020|mr=1707327}}A.Scorpan, The wild world of 4-manifolds (p.90), AMS Pub. {{isbn|0-8218-3749-4}}

A compact contractible Stein 4-manifold $C$ with involution $F$ on its boundary is called an Akbulut cork, if $F$ extends to a self-homeomorphism but cannot extend to a self-diffeomorphism inside (hence a cork is an exotic copy of itself relative to its boundary). A cork $(C,F)$ is called a cork of a smooth 4-manifold $X$, if removing $C$ from $X$ and re-gluing it via $F$ changes the smooth structure of $X$ (this operation is called "cork twisting"). Any exotic copy $X\text{'}$ of a closed simply connected 4-manifold $X$ differs from $X$ by a single cork twist.{{cite journal|last= Akbulut|first=Selman|author-link=Selman Akbulut|title= A fake compact contractible 4-manifold|journal= Journal of Differential Geometry|volume= 33|issue=2|year=1991|pages=335–356|mr=1094459|doi=10.4310/jdg/1214446320|doi-access=free}}{{cite journal|first=Rostislav| last=Matveyev|title= A decomposition of smooth simply-connected h-cobordant 4-manifolds|journal= Journal of Differential Geometry|volume= 44 |year=1996|issue= 3|pages= 571–582|mr=1431006|doi=10.4310/jdg/1214459222|arxiv=dg-ga/9505001| s2cid=15994704}}{{cite journal|first1=Cynthia L.|last1= Curtis|first2= Michael H.|last2= Freedman|author2-link=Michael Freedman| first3=Wu Chung|last3= Hsiang|author3-link=Wu-Chung Hsiang|first4=Richard|last4= Stong|title= A decomposition theorem for h-cobordant smooth simply-connected compact 4-manifolds|journal=Inventiones Mathematicae|volume= 123 |year=1996|issue= 2|pages=343–348|mr=1374205|doi=10.1007/s002220050031|s2cid= 189819783}}{{cite journal|last1= Akbulut|first1=Selman|author1-link=Selman Akbulut|first2=Rostislav| last2=Matveyev| title=A convex decomposition theorem for 4-manifolds|journal= International Mathematics Research Notices |year=1998|volume=1998 | issue= 7|pages= 371–381|mr=1623402|doi=10.1155/S1073792898000245 |doi-access=}}{{cite journal|last1= Akbulut|first1=Selman|author1-link=Selman Akbulut|first2=Kouichi| last2=Yasui| title=Corks, plugs and exotic structures|url=http://gokovagt.org/journal/2008/jggt08-akbulutyasui.pdf| journal= Journal of Gökova Geometry Topology |volume= 2 |year=2008|pages= 40–82|arxiv=0806.3010 |mr=2466001}}

The basic idea of the Akbulut cork is that when attempting to use the h-cobordism theorem in four dimensions, the cork is the sub-cobordism that contains all the exotic properties of the spaces connected with the cobordism, and when removed the two spaces become trivially h-cobordant and smooth. This shows that in four dimensions, although the theorem does not tell us that two manifolds are diffeomorphic (only homeomorphic), they are "not far" from being diffeomorphic.Asselmeyer-Maluga and Brans, 2007, Exotic Smoothness and Physics

To illustrate this (without proof), consider a smooth h-cobordism $W^5$ between two 4-manifolds $M$ and $N$. Then within $W$ there is a sub-cobordism $K^5$ between $A^4\; \backslash subset\; M$ and $B^4\; \backslash subset\; N$ and there is a diffeomorphism

:$W\; \backslash setminus\; \backslash operatorname\{int\}\backslash ,\; K\; \backslash cong\; \backslash left(M\; \backslash setminus\; \backslash operatorname\{int\}\backslash ,\; A\; \backslash right)\; \backslash times\; \backslash left[0,1\backslash right],$

which is the content of the h-cobordism theorem for n ≥ 5 (here int X refers to the interior of a manifold X). In addition, A and B are diffeomorphic with a diffeomorphism that is an involution on the boundary ∂A = ∂B.Scorpan, A., 2005 The Wild World of 4-Manifolds Therefore, it can be seen that the h-corbordism K connects A with its "inverted" image B. This submanifold A is the Akbulut cork.