---
abstract: 'We show that the smooth $4$-manifold $M$ obtained by attaching a $2$-handle to $B^4$ along a certain knot $K\subset \partial B^4$ admits infinitely many absolutely exotic copies $M_n$, $n=0,1,2..$, such that each copy $M_n$ is obtained by attaching $2$-handle to a fixed compact smooth contractible manifold $W$ along the iterates $f^{n}(c)$ of a knot $c\subset \partial W$ by a diffeomorphism $f:\partial W \to \partial W$. This generalizes the example in author’s 1991 paper, which corresponds to $n=1$ case.' 
address: 'Department of Mathematics, Michigan State University, MI, 48824'
author:
- Selman Akbulut
date:
- 
- 
title: A simple class of infinitely many absolutely exotic manifolds
---

5.9in 7.9in

.75in .75in

\[Thm\][Lemma]{} \[Thm\][Corollary]{} \[Thm\][Proposition]{}

¶[[P]{}]{} §[[S]{}]{} Ø[[O]{}]{} [H]{}

[^1]

Construction
============

A relative exotic structure on a compact smooth $4$-manifold $M$ with boundary, is a self diffeomorphism $f:\partial M\to \partial M$, which extends to a self homeomorphism of $M$, but does not extend to a self diffeomorphism of $M$. If $F:M\to M$ a homeomorphism extending $f$, then the pull-back smooth structure $M_{F}$ provides a relative exotic copy of $M$. We say that $N$ is an absolutely exotic copy of $M$, if it is homeomorphic but not diffeomorphic to $M$ (no condition on the boundary).  The technique introduced in [@ar] turns relative exotic structures to absolute exotic structures.  This is done by choosing an invertable cobordism $H$ with $\partial H=H_{-}\sqcup H_{+}$ and $H_{-}\approx \partial M$, and then gluing $H$ it to the boundary of $M$ in two different ways.  Then the manifolds $M'=M\cup_{Id}H$ and $M''=M\cup_{f}H$ become absolutely exotic copies of each other.  Applying this construction to a cork $W$ produces an absolutely exotic copy of $W$; and when $W$ is an infinite order loose-cork ([@a2]) then we get infinitely many absolutely exotic copies of $W$. This construction results a boundary $H_{+}$, which consists of hyperbolic manifolds glued along tori. 

Ideally we want to produce small $4$-manifolds with simple boundaries, admitting absolutely exotic copies (with the hope of capping boundaries to get small closed exotic manifolds).  One such example is the cusp $C^4$ ([@a3]) with a Seifert fibered space boundary, which is obtained by attaching a $2$-handle to $B^4$ along the trefoil knot with $0$-framing.  Performing knot surgeries $C\leadsto C_{K}$ along the torus inside (by using different knots $K$) provides infinitely many absolutely exotic copies of $C$.  An interesting open problem here is to find corks inside of $C$, such that twisting $C$ along them induce the exotic copies $C_{K}$ of $C$.  Here we produce another example $M^4$, similar to $C^4$, which also has a Seifert fibered space boundary, and is obtained by attaching a $2$-handle to $B^4$ along a slice knot with $-2$ framing.  But from this we can construct infinitely many different exotic copies of M, each obtained by cork-twisting along an infinite order loose-cork $W\subset M$ ([@a2]), rather than a knot surgery to $M$, as in the case of $C$ above. 

\[thm1\] The manifold $M$ of Figure \[a1\], which is obtained by attaching $2$-handle to $B^4$ along the knot $K$ of Figure \[a1\] with $-2$ framing, admits infinitely many distinct absolutely exotic copies, and they can be detected twisting an infinite order loose-cork inside of $M$.

![ M[]{data-label="a1"}](a1.eps){width=".19\textwidth"}

First of all by blowing up and down as in Figure \[a2\], we see that $\partial M$ can also be identified by $+2$ surgery to $(-4, 2)$ twist knot (stevedores knot).  $\partial M$ is the small Seifert fibered space $M (-2;1/2,3/4,7/9)$ (e.g. [@bw], [@s], [@t]), therefore its mapping class group is finite order. 

![ M[]{data-label="a2"}](a2.eps){width=".5\textwidth"}

Now, recall the infinite order loose-cork $W$ of [@a2], which is shown in Figure \[a3\] (where two of its alternative handlebody pictures are given).  The order $n$ diffeomorphism $f_{n}: \partial W \to \partial W$ is obtained by a delta move along the curve $\delta \subset \partial W$ (the dotted curve in the figure). 

![ W[]{data-label="a3"}](a3.eps){width=".7\textwidth"}

Now check that the handlebody pictures of Figure \[a4\] describe the manifold $M$ above (to see this cancel $1/2$- handle pairs).  The second picture of Figure \[a4\] shows an imbedding $W\subset M$. That is, $M$ is obtained from $W$ by attaching a $2$-handle along the knot $c$ with $0$-framing. 

![ M[]{data-label="a4"}](a4.eps){width=".75\textwidth"}

Now apply $\delta$- moves to $W$ inside $M$, $n$-times (where $\delta$ is chosen as in Figure \[a3\]), and call the resulting manifold $M_{n}$ (which is the first picture of Figure \[a5\]).  We claim $\{M_{n}\}$ are exotic copies of $M$ rel boundary.  To see this attach $2$-handles to $M_{n}$ along the knots $a$ and $b$ of the picture. 

![ $M_{n} \leadsto S_{n}=M_{n}+a^{-1}+b^{-1}$[]{data-label="a5"}](a5.eps){width=".7\textwidth"}

Call the manifold obtained from $M_{n}$ by attaching $2$-handles along $a$ and $b$ with $-1$ framings by $S_{n}= M_{n} +a^{-1}+b^{-1}$. Now we proceed as in [@a2] by handle slides, to show that $S_{n}$ is the manifold obtained from the Stein manifold $S$ of Figure \[a7\] by the knot surgery using the twist knot $(-2, -n)$. Furthermore we can compactify $S$ into some closed symplectic manifold $Z$ with $b_{2}^{+}(Z)>1$ (by [@lm], [@ao], or [@a4] p.108). 

![ $S_{n}$[]{data-label="a6"}](a6.eps){width=".7\textwidth"}

![ S[]{data-label="a7"}](a7.eps){width=".6\textwidth"}

This shows that manifolds $\{M_{n}\}$ are exotic copies of $M$ rel boundary, and they are obtained by iterating $\delta$-moves to $f: \partial W \to \partial W $ inside $W\subset M$. Since the mapping class group of the Seifert fibered space $\partial M \approx M_{n}$ is finite [@bo], by going to a subsequence we can assume that all $\{M_{n}\}$ are absolutely exotic copies of each other. 

Now we analyze what an $n$-iterate of the $\delta$- move does to $M$, well it turns it into $M_{n}$, and a close inspection shows that $M_{n}$ is obtained from $M$ by attaching $2$-handle to $W$ along the loop $f^{n}(c)$ with $0$-framing, as shown in Figure \[a8\].  Recall also, performing the $\delta$-move to $W$ inside of $M$, has the affect of attaching a cancelling pair of $2/3$-handles to $M$ and performing the diffeomorphism described in [@a1] resulting $M_{n}$.

![ $M_n$[]{data-label="a8"}](a8){width=".7\textwidth"}

Note the new features of Theorem \[thm1\] which can not be reached by the techniques of [@ar], they are: (1) We don’t need to modify the boundary of $M$ (by a homology cobordism) in order to construct its absolutely exotic copy. ( 2) The construction here produces infinitely many absolutely exotic copies of $M$. (3) $M$ contains the tangent disc bundle of $S^2$ (an imbedded $-2$ sphere) and vice versa, so every smooth manifold which contains a $-2$ sphere contains a copy of $M$ inside. 

[Acknowledgements: I thank Danny Ruberman for making suggestions and discussing some of the constructions of this paper with me.]{} 

[99999]{}

S. Akbulut, [*An exotic 4-manifold*]{}, J. Differ.  Geom.  33 (1991) 357–361. 

S. Akbulut, [*On infinite order corks*]{}, PGGT 2016

S. Akbulut, [*A fake cusp and a fishtail.*]{}  Turkish J. Math.  1 (1999) 19–31. 

S. Akbulut [*$4$-Manifolds*]{}, Oxford Graduate Texts in mathematics (2016) ISBN: 9780198784869

S. Akbulut and B. Ozbagci [*On the topology of compact Stein surfaces* ]{}, Int.  Math.  Res.  Notices 15 (2002) 769-782

S. Akbulut and D.Ruberman [*An absolutely exotic contractible 4-manifold.*]{}  Commentarii Mathematici Helvetici, (2016) 91(1) 1-19

M. Boileau and J.-P. Otal [*Scindements de Heegaard et groupe des homŽotopies des petites variŽtŽs de Seifert.*]{}  Invent.  Math.106 (1991), vol 1, 85 -107. 

Mark Brittenham and Ying-Qing Wu [ *The classification of exceptional Dehn surgeries on $2$-bridge knots*]{}.  Comm.  Anal.  Geom.  9 (2001), vol1, 97-113. 

N. Saveliev [*Invariants for homology $3$-spheres*]{}, Encylopaedia of Math.  Sci vol.  140, Springer- Verlag ISBN 3-540-43796-7

P. Lisca and G. Mattic [*Tight contact structures and Seiberg-Witten invariants,*]{} Invent.  Math.  129 (1997) 509-525. 

S. P. Plotnick [*Vanishing of Whitehead groups of Seifert manifolds with infinite fundamental groups*]{}, Comm.  Math.  Helv.  55 (1980) 654-667. 

B. Tosun [*Tight small Seifert fibered manifolds with $e_{0}=-2$*]{},\
https://arxiv.org/pdf/1510.06948.pdf

[^1]: Partially supported by NSF grants DMS 0905917
