Journal of Gökova Geometry Topology
Volume 9 (2015) 1 – 17

A plug with infinite order and some exotic 4-manifolds
Motoo Tange
Abstract. It is known that a closed, orientable, simply-connected 4-dimensional
manifold X ′ which is an exotic copy of X can be obtained by removing a submanifold
C (called a cork) of X and regluing C back. We can assume that C is contractible
and the gluing map is an involution. In this paper we define corks and plugs withorder p ∈ N≥2 ∪ {∞} and we show a plug (P, ϕ) with infinite order which produces“a crossing change” of Fintushel-Stern’s knot surgery where (P, ϕ2 ) is a (generalized)
cork with infinite order.

1. Introduction
1.1. Smooth structures
Let X, X ′ be two smooth manifolds. If X ′ is homeomorphic but non-diffeomorphic to
X, then we say that X and X ′ are exotic (or exotic pair). Here we define cork and plug
according to [AY1].
Definition 1.1 (Cork). Let C be a compact, contractible, Stein 4-manifold with boundary and τ : ∂C → ∂C a diffeomorphism with τ 2 = id. (C, τ ) is called a cork, if the mapτ cannot extend to any diffeomorphism of C.
We call (C, τ ) a cork of a 4-manifold X, if there exists an embedding i : C ֒→ X
such that when removing C ⊂ X and regluing C via the diffeomorphism τ , the resulting
manifold X ′ = (X − C) ∪τ C is exotic to X. This deformation X
X ′ is called a cork
twist and presented as (C, τ, i).
The cut and paste notation like (X − Z) ∪ϕ Y will be defined in Section 1.2. Whenthe embedding i of the cork C is clear from the context, the cork twist is denoted simply
by (C, τ ).
As proven in [AM, CFHS, M], any closed, simply-connected exotic pair can be obtainedby a cork twist (C, τ, i). Thus, how to find corks for every exotic pair is a next naturalproblem.
Akbulut and Yasui in [AY1] introduced another type of a construction (P, τ ) to givean exotic pair.
Key words and phrases. 4-manifold, exotic structure, cork, plug, Fintushel-Stern’s knot-surgery.
The work was supported by Grant-in-Aid for JSPS Fellows 21-1485 and JSPS KAKENHI Grant
Number 24-840006.

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Definition 1.2 (Plug). Let (P, τ ) be a compact, Stein 4-manifold P with boundary andτ : ∂P → ∂P a diffeomorphism with τ 2 = id. We assume that τ cannot be extended to ahomeomorphism P → P . We call (P, τ ) a plug if there exist a smooth 4-manifold X andan embedding i : P ֒→ X such that when removing P ⊂ X and regluing P via the map
τ , the resulting manifold X ′ = (X − P ) ∪τ P is exotic to X. This deformation X
X′
is called a plug twist and presented as (P, τ, i).
Recently many smooth structures have been constructed by using corks and plugs as in[A2, AY1, AY2, AY3, AY4, GS]. Since the main idea for the existence of cork is due to thefailure of smooth 4-dimensional h-cobordism theorem, naturally the self-diffeomorphismof the boundary of corks and plugs must be an involution. For example, as appearedin [A1, BG], a cork C for an exotic pair E(2)#CP 2 and #3 CP 2 #20 CP 2 is given by a
Mazur type 4-manifold with a symmetric framed link diagram. The involution is the one
induced by the symmetry of the framed link. Here E(n) is the elliptic fibration with 12n
Lefschetz singularities.
Let T ⊂ X be an embedded torus with the trivial normal bundle. Let K be a knot in
S 3 . MK is the 0-surgery along K and Tm is a torus m × S 1 in MK × S 1 , where m is themeridian of K. (Fintushel-Stern’s) knot-surgery [FS1] is defined to be
XK = (MK × S 1 )#Tm =T X.
See Section 1.2 for the notation for the fiber-sum.
We give a brief review about the (2-component) link-surgery in [FS1]. Let U1 , U2 be
two 4-manifolds and Ti ⊂ Ui embedded tori with trivial neighborhoods. Let L = K1 ∪ K2
be a 2-component link. Let
αL : π1 (S 3 − L) → Z
be a homomorphism satisfying αL (mi ) = 1, where mi is the meridian curve of Ki . Let
ML be the α(ℓi )-surgery of L, where ℓi is the longitude of Ki . Let U = (U1 , U2 ) denote acollection of 4-manifolds Ui and the embedded tori Ti , and Tmi = mi × S 1 , where mi are
the meridians of Ki in ML . Then, we denote by UL the following:
UL = U1 #T1 =Tm1 (ML × S 1 )#Tm2 =T2 U2 .
In the case of U = U1 = U2 , we write as UL = UL . We call UL the link-surgery by the link
L. This link-surgery operation can be generalized to an n-component link version.
Our motivation is to construct “a cork (or plug)” representing the knot-surgery or
link-surgery. Since the knot-surgery gives infinitely many exotic structures, we, naturally,
consider corks and plugs with infinite order. In the next subsection we will define suchcorks and plugs with order p, where p is greater than or equal to 2 or ∞.
One of the main aims in this paper is to give an example of a (generalized) cork with
infinite order. It is the square (P, ϕ2 ) of a plug (P, ϕ) with infinite order. Let X bea 4-manifold as described in Figure 1. This manifold is diffeomorphic to a neighborhood of Kodaira’s singular fiber with type III (see Table 1.9 in [HKK]). We denote theneighborhood of the fiber by V .
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A plug with infinite order and some exotic 4-manifolds

Figure 1. Kodaira’s singular fibration V of type III
Remark 1.3. Throughout this paper, unlabeled links in the handle diagrams of
4-manifolds or 3-manifolds stand for 0-framed 2-handles or 0-surgeries respectively.
Remark 1.4. The handle decompositions of knot-surgery and link-surgery are based on
the descriptions in [A3, T1, T2].
Let P be the simply-connected 4-manifold with the handle decomposition given in
Figure 2. We will define a diffeomorphism ϕ : ∂P → ∂P in Section 3.
Theorem 1.5. The (P, ϕ)-twist is a plug with infinite order.
This plug (P, ϕ) gives in some sense a crossing change concerning Fintushel-Stern’sknot-surgery.
Theorem 1.6. The (P, ϕ2 )-twist is a generalized cork.
This meas the (P, ϕ)-twist cannot extend to the inside of P as a homeomorphism, but
the square can extend to the inside of P as a homeomorphism.
In general we show the following theorem.
Theorem 1.7. Let X be a 4-manifold containing V and K a knot in S 3 . Let XK be theknot-surgery along the general torus fiber in V . Suppose that K ′ is one crossing changeof K. Then there exists an embedding P ֒→ XK such that the plug twist of XK is XK ′ .
Let K n be an n-times crossing change of a knot K. Then the infinitely many exotic
4-manifolds XK n are obtained by iterating the plug twist (P, ϕ).
Variations and application of the plug (P, ϕ) can be seen in the forthcoming paper[T3].

1.2. Notation on cut-and-paste.
Here we define a notation on cut-and-paste operations on manifolds.
Definition 1.8 (Cut-and-paste). Let X be an n-manifold and i : Z ֒→ X an embeddingof compact, codimension-0 submanifold Z with boundary. Here the boundary ∂Z = Σ
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is a smooth (n − 1)-manifold. Let Y be an n-manifold with a fixed diffeomorphism map
Σ → ∂Y . Let ϕ be a map Σ → Σ. The manifold obtained by identifying ∂(X − Z) and∂Y via the map ϕ is denoted as follows:(X − Z) ∪ϕ Y.
In particular, if Y = Z, then the surgery (X − Z) ∪ϕ Z is called (Z, ϕ, i)-twist of X. If
the embedding i is clear from the context, we call the twist a (Z, ϕ)-twist.
Let X, Y be 4-manifolds with tori T1 ⊂ X, T2 ⊂ Y with trivial neighborhoods. Then,the resulting manifold by doing the fiber-sum of X and Y as defined in [GS] is written as
follows:
X#T1 =T2 Y := (X − ν(T1 )) ∪ϕ (Y − ν(T2 )),where ν(·) stands for the neighborhood and ϕ is a fiber-preserving diffeomorphism. In
fact, the diffeomorphism type depends on the diffeomorphism ϕ.

2. Cork and Plug with order p.
In Section 1.1 we gave the original definition of corks and plugs. Here we define notionsof corks and plugs with order p.
Definition 2.1 (Cork with order p). Let (C, ϕ) be a pair of a compact, contractible,
Stein 4-manifold C with boundary and a diffeomorphism ϕ : ∂C → ∂C. Let p be an
integer with p ≥ 2 or infinity. The pair (C, ϕ) is called a cork with order p if it satisfies
the following properties:(1) The order of ϕ is p.
(2) For any 0 < k < p the map ϕk cannot be extended to ϕk : C → C as a diffeomorphism.
If there exist p mutually exotic 4-manifolds X = {X, X1 , X2 , · · · , Xp−1 } (in the caseof p < ∞) or X = {X, X1 , X2 , · · · } (in the case of p = ∞) such that an embeddingi : C ֒→ X and a diffeomorphism ϕ : ∂C → ∂C produce the 4-manifolds
Xk = [X − C] ∪ϕk C,then (C, ϕ, i) is a cork with order p of X .
We also treat a non-contractible cork C as appeared in [AY1]. It is called a generalized cork with order p. The regluing map ϕ for any generalized cork must extend to ahomeomorphism C → C.
Definition 2.2 (Plug with order p). Let (P, ϕ) be a pair of a compact Stein 4-manifold
P with boundary and a diffeomorphism ϕ : ∂P → ∂P . Let p be an integer with p ≥ 2 orinfinity. The pair (P, ϕ) is called a plug with order p if it satisfies the following properties:(1) The order of ϕ is p.
(2) The map ϕ cannot be extended to any self-homeomorphism of P .
4

A plug with infinite order and some exotic 4-manifolds(3) There exists a 4-manifold X containing P such that for any 0 < k < p, X and
Xk = [X − P ] ∪ϕk P
are p mutually exotic manifolds.
The classical cork or plug corresponds to the p = 2 case in the definition above.
Although we give an example of an infinite order plug, it is unknown whether there exists
a finite order plug with p > 2 or not.
Question 2.3. Does there exist any plug or cork (Q, ψ) with finite order p (3 ≤ p < ∞)?

3. A plug (P, ϕ) with infinite order.
We construct a plug (P, ϕ) with infinite order. We verify that the pair satisfies all of
the properties (1), (2), and (3) for (P, ϕ) to be a plug with infinite order.

3.1. The definition of P and the topological invariants.
Let P be the compact 4-manifold admitting the handle decomposition as in Figure 2.
The manifold P is also obtained by attaching three −1-framed 2-handles on V2 × S 1 ,Figure 2. A handle decomposition of P .

Figure 3. The generatorsin π1 (∂P ).

where V2 is the genus 2 handlebody. Since the three 2-handles kill the three generatorsin the π1 (V2 × S 1 ), P is simply-connected. Thus, we have H2 (P ) ∼= Z2 . The presentationof π1 (∂P ) is the following:
π1 (∂P )=ha, b, c, d, f |[f, b−1 ], [f, a−1 ], a[f, c−1 ], b[f, d−1 ],f −1 [c−1 , f ][c, f ][d−1 , f ][d, f ]i,∼=

hc, d, f |[f, [f, d−1 ]], [f, [f, c−1 ]], f −1 [c−1 , f ][c, f ][d−1 , f ][d, f ]i,where we use generators a, b, c, d and f in Figure 3 and the symbol [x, y] stands for thecommutator xyx−1 y −1 . Therefore we get H1 (∂P ) ∼= Z2 .
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3.2. A Stein structure on P .
In order for a 4-manifold X to admit a Stein structure, we must show that it has a
specific handle decomposition. That is, as in [E] or [GS], X is a 2-handlebody, and theattaching spheres of 2-handles have Legendrian presentation in #n S 2 ×S 1 = ∂(♮n D3 ×S 1 )such that each framing is tb − 1, where tb stands for the Thurston-Bennequin invariant of
the Legendrian knot. To show that the 4-manifold P is a plug we must show that P is acompact Stein manifold. Canceling the two 1-handles, P admits the handle decomposition
as in Figure 4. The Thurston-Bennequin invariants of the components are all 1. Hence,
this handle decomposition satisfies Eliashberg’s condition above. Therefore we get thefollowing proposition:
Proposition 3.1. P admits a Stein structure.

Figure 4. A Stein structure over P .

3.3. A diffeomorphism ϕ on ∂P .
We define a diffeomorphism ϕ on ∂P . Sliding and isotopy in Figure 5 gives a diffeomorphism on ∂P and we call it ϕ. Hence the generators c, d and f in π1 (∂P ) are mapped
via ϕ to the following respectively:(Xb)−1 a(Xb)a−1 c(Xb)
d(Xd)−1 a(Xd)and f,
where X = [b

−1

, d].

Lemma 3.2. The map ϕ acts on H1 (∂P ) and H2 (∂P ) trivially.
Proof. Taking the abelianization of the images of c, d and f via ϕ, we see that the mapϕ∗ : H1 (∂P ) → H1 (∂P ) is the identity. Hence, through the Poincaré dual, the action on
H2 is also the identity.

The map ϕ trivially acts on the homology, but the order of ϕ is infinite. This will be
proven in the next subsection.
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A plug with infinite order and some exotic 4-manifolds

Figure 5. The diffeomorphism ϕ.

3.4. Infinite exotic structures which (P, ϕi )-twists give.
In this subsection we show that there exists a compact 4-manifold X containing P suchthat infinitely many surgeries(X − P ) ∪ϕi P
give infinitely many exotic structures on X. This is the most essential part among theproperties of this plug. As a corollary, ϕ has infinite order in the mapping class group on∂P .
Proposition 3.3. Let Kn be a twist knot in S 3 given by the diagram in Figure 6. Let
V be a neighborhood of singular fiber of Kodaira’s type III and VKn the knot-surgery of
V . Then, there exists P ֒→ VKn (n ≥ 0) such that the resulting manifolds(VKn − P ) ∪ϕ P = VKn+1 ,7

TANGE(V − P ) ∪ϕn P = VKn
and the order of ϕ is infinite. In particular, {VKn } contains infinitely many exotic copies
of V .

Figure 6. The diagram of Kn , where the box with n in the picture
represents n full twists.

Proof. Let Kn be the 2n-twist knot as in Figure 6. The neighborhood V is the left of
Figure 7. Performing Fintushel-Stern’s knot-surgery [FS1] on V , we get the deformation

Figure 7. Fintushel-Stern’s knot-surgery of X.

in Figure 7. Sliding one of the two −1-framed 2-handles over 0-framed 2-handle in the
diagram, and removing the top 0-framed 2-handle, we get the handle diagram of P in
VKn (see Figure 8).
Here, we perform the (P, ϕ)-twist of VKn along P as follows:(VKn − P ) ∪ϕ P.
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A plug with infinite order and some exotic 4-manifolds

Figure 8. The submanifold P ⊂ VKn .
We keep track of the handle moves in Figure 5 to VKn . In Figure 9, any attachingspheres of 3-handles are not described. However, due to [Tr], via the (P, ϕ)-twist, theextension to the 3-handles are unique.
As a result, it turns out that the surgery adds the full twist to the diagram of VKn (seeFigure 9). Consequently, we get
VKn → (VKn − P ) ∪ϕ P = VKn+1 .

Figure 9. The (P, ϕ)-twist gives VKn → (VKn − P ) ∪ϕ P = VKn+1 .

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Figure 10 determines a diffeomorphism φn : ∂V → ∂VKn . More precisely, Figure 10
itself gives a diffeomorphism on ∂V #S 2 × S 1 , however the part of S 2 × S 1 does notchange. For, ∂V is a prime manifold, because ∂V is a torus bundle over S 1 and theuniversal cover is R3 , in particular, π2 (∂V ) = 0. Further, it is well-known that the
separating sphere of the connected-sum of two prime 3-manifolds is unique up to isotopy.
Thus any diffeomorphism ∂V #S 2 × S 1 gives a diffeomorphism on ∂V uniquely.
VKn+1 above induces the boundary diffeomorphism φn+1 ◦ φ−1
The (P, ϕ)-twist VKnnbetween ∂VKn and ∂VKn+1 . The diffeomorphism is the identity map via the diffeomorphisms φn , φn+1 . Note that in the diffeomorphism, the same handle move process as ϕup to isotopy.
Actually, φn : ∂V → ∂VKn is the same map as the boundary diffeomorphism fromknot-surgery, as in Figure 5 in [A3]. Thus, we can get the result by (P, ϕ)-twist as
follows:
E(2)Kn → (E(2)Kn − P ) ∪ϕ P ∼= E(2)Kn+1 .
It is well-known that the knot-surgery E(2)K gives exotic structures. The SeibergWitten invariant formula on Fintushel-Stern’s knot-surgery in [FS1] says that:
SWE(2)Kn = −nt + 2n + 1 − nt−1 .
This implies that E(2)Kn are infinitely many exotic 4-manifolds. Thus, the (P, ϕi )-twist ofE(2) = E(2)K0 gives rise to infinitely many mutually exotic 4-manifolds {E(2)Ki |i ∈ Z}.
Hence, X = E(2) = E(2)K0 satisfies this proposition. In particular, the map ϕ has
infinite order.
As a corollary we get the following:
Corollary 3.4. Let X be a 4-manifold containing V . Let Kn be a twist knot as in
Figure 6. Then, the plug twist (P, ϕ) of VKn ⊂ XKn produces XKn+1 .
Proof. The (P, ϕ)-twist of VKn gives VKn+1 . The diffeomorphism ∂VKn → ∂VKn+1 via thetwist is the same map as the gluing map of the knot-surgery as is presented in Figure 5
of [A3]. Thus, the manifold XKn is changed to XKn+1 by (P, ϕ)-twist.

3.5. Extendability as a homeomorphism.
Lastly we show that ϕ cannot be extended to a self-homeomorphism on P . To provethat, we use Boyer’s theorem in [B]. First, we prepare with the following proposition.
Proposition 3.5. The twisted doubles Dϕn (P ) := P ∪ϕn P have the following properties:
spin: n evenDϕn (P ) is
not-spin : n odd.
Since any double Dϕn (P ) is simply-connected, we may see whether the intersection
form QDϕn (P ) is even or odd. Note that the intersection form of P is even.

10A plug with infinite order and some exotic 4-manifolds

Figure 10. The diffeomorphism φn : ∂V → ∂VKn .

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Proof. First the double D(P ) := P ∪id P is diffeomorphic to #2 S 2 × S 2 by the handle
calculus in Figure 11. The twisted double Dϕ (P ) is diffeomorphic to #2 (CP 2 #CP 2 ) as
in Figure 12.
In general, Dϕn (P ) is diffeomorphic to the manifold given by the left diagram in
Figure 13. The change of the framing of the bottom 2-handle in this picture to the
0-framing gives the manifold Dn (the right of Figure 13). Since Dn is the diagram ofsome closed 4-manifold, the change is a kind of homeomorphic deformation. Furthermore,
the application of the local move in Figure 14 to the Dn yields Figure 15. Namely, we
get the classification:(#2 (CP 2 #CP 2 ) if n odd∼
Dn =#2 S 2 × S 2
if n even.
Therefore, Dϕn (P ) is spin or non-spin for n is even or odd respectively.

Figure 11. D(P ) = P ∪id P .
The problem of whether Dϕn (P ) is diffeomorphic to Dn or not is remaining.
Question 3.6. Is Dϕn (P ) for |n| ≥ 2 an exotic #2 S 2 × S 2 or #2 (CP 2 #CP 2 )?
Applying Proposition 3.5 to (iii) in (0.8) Proposition in Boyer’s paper [B], we get thefollowing:
Corollary 3.7. For any integer n the diffeomorphism ϕn : ∂P → ∂Pcannot be extended to a self-homeomorphism on P : n odd
can be extended to a self-homeomorphism on P
: n even.
In other words the following holds:
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A plug with infinite order and some exotic 4-manifolds

Figure 12. Dϕ (P ) = P ∪ϕ P .

Figure 13. Dϕn (P ) → Dn (homeomorphism).
Corollary 3.8. For each non-zero integer m, each (P, ϕ2m )-twist is a generalized corkwith infinite order and each (P, ϕ2m+1 ) is a plug with infinite order.
In particular, this corollary proves Theorem 1.6.
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Figure 14. A local move which gives a diffeomorphism.

Figure 15. Diffeomorphism types of Dn .

Proof of Theorem 1.5.
We show the (P, ϕ)-twist satisfies the condition of a plug with infinite order. Proposition 3.1 shows P is a Stein manifold. Proposition 3.3 shows ϕ has infinite order and(P, ϕ)-twists produce infinitely many exotic manifolds. Corollary 3.7 shows that ϕ cannot
extend inside P .
Proof of Theorem 1.7.
Let D be a diagram of a knot K in S 3 . Given a crossing point in D as in Figure 16,
we move the point in the bottom position in the picture by isotopy. Second, we movethe local minimum points and local maximal points in the bottom and top positions
respectively. The resulting picture is a plate presentation of K that the crossing point isa bottom position in the braid. This presentation can give the handle decomposition of[S 3 − ν(K)] × S 1 . For the way to describe any knot-surgery and [S 3 − ν(K)] × S 1 one cansee [A3, T2]. VK is [S 3 − ν(K)] × S 1 attaching the 3 vanishing cycles. See Figure 17 for
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A plug with infinite order and some exotic 4-manifolds

the embedding P ֒→ VK . Applying the (P, ϕ)-twist (the process in Figure 5) to P ⊂ VK ,we obtain VK ′ , where K ′ is the crossing change of K.
We show that the (P, ϕ)-twist deforms XK into XK ′ . We give φK : ∂V → ∂VK in thesimilar way to Figure 10. The map φK is defined by using a plate presentation of K,
however this is independent of the plate presentation of K. Suppose that φK and φ′K are
two boundary diffeomorphisms for different presentations. Here there exists an isotopy
from ∂VK to ∂VK such that φK (b) and φK (c) are isotopic to φ′K (b) and φ′K (c), because
we can make an isotopy between two presentations with φK (b) and φK (c) fixed. Thus′−1
φ′−1
K ◦ φK (b) = b and φK ◦ φK (c) = c hold up to isotopy.
The maps φK and φ′K induces two fibrations π, π ′ : ∂VK → S 1 respectively comingfrom ∂V → S 1 . As mentioned above, φ′−1
K ◦ φK is a bundle isomorphism ∂V → ∂V fixing
the fibers. Here φ′−1◦
φinducesadiffeomorphism
S 1 → S 1 it is isotopic to identity.
KK
1
The mapping class group of S is isomorphic to Z/2Z and the isotopy class is determinedby the change of the orientation of S 1 . Actually, the linking number of the curve a in
Figure 10 with the circle representing the 0-framed 2-handles is preserved for φK and′−111φ′K . Thus, the map φ′−1
K ◦ φK is reduced to the identity S → S . Therefore, φK ◦ φK′is the identity map up to isotopy. In particular, φK = φK holds up to isotopy.
∂V
π0


S1

φK

/ ∂VK oπ π′/ S1 o

φ′K

∂V
π0


S1

This diffeomorphism gives a diffeomorphism on ∂V #S 2 × S 1 # · · · #S 2 × S 1 . Any diffeomorphism of ∂V #S 2 × S 1 # · · · #S 2 × S 1 can give a diffeomorphism on ∂V uniquely,
because any attachment of the 3-handles in VK is unique (due to [Tr]). In particular, thesphere separating as ∂V and S 2 ×S 1 # · · · #S 2 ×S 1 is uniquely determined. Consequently,
this φK does not depend on the way of the plate presentation of K.
Therefore, φK ◦ φ−1
K ′ : ∂VK ′ → ∂VK is the same handle move as that of ϕ. The
diffeomorphism ∂V → ∂VK is the diffeomorphism induced by knot-surgery. See Figure 5
in [A3]. This implies XK → XK ′ .


Remark 3.9. Let L1 , L2 be two unknotting number 1 knots. Then there exist embedE(2)L1
dings ιL1 , ιL2 : P ֒→ E(2) such that the plug twists give the knot-surgery E(2)and E(2)E(2)L2 . Therefore the embeddings ιL1 and ιL2 are not smoothly isotopic
each other, if ∆L1 6= ∆L2 .
Remark 3.10. Let X be a manifold containing V . Note that the knot-surgery XKn
along a general fiber of V is the plug twist (P, ϕn , i) for odd n. On the other hand, since
Kn is an unknotting number 1 knot, there exists another plug twist of X by (P, ϕ−1 , κn )15

TANGEa plate presentation of K
Figure 16. A presentation of any knot with a given crossing in a bottom position.

Figure 17. P ֒→ VK

such that the plug twist gives XKn .
(P,ϕ−1 ,κn )

/ XK
X ◆◆
n
◆◆◆
◆◆◆
◆◆
(P,ϕn ,i) ◆◆◆'XK n

4. Some questions.
Here we summarize the several questions.
Question 2.3. Does there exist any plug or cork (Q, ψ) with finite order p (3 ≤ p < ∞)?
Question 3.6. Is Dϕn (P ) for |n| ≥ 2 an exotic #2 S 2 × S 2 or #2 (CP 2 #CP 2 )?
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A plug with infinite order and some exotic 4-manifolds

Acknowledgements: I thank Kouichi Yasui and Yuichi Yamada for giving me some
useful suggestions and advice to study plugs with infinite order and also thank Mikio
Furuta for a useful comment in the meeting “Four-Dimensional Topology in 2011” at
Hiroshima University.

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