FINITE ORDER CORKS
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Abstract. We show that for any positive integer m, there exist order n Stein corks. The boundaries are cyclic branched covers of slice
knots embedded in the boundary of a cork. By applying these corks
to generalized forms, we give a method producing examples of many
finite order corks, which are possibly not Stein cork. The examples of
the Stein corks give n homotopic and contactomorphic but non-isotopic
Stein filling contact structures for any n.

1. Introduction
1.1. 4-manifold and cork. If two smooth manifolds X1 and X2 are homeomorphic but non-diﬀeomorphic, then we say that X1 and X2 are exotic.
Let X be a smooth manifold and Y a codimension 0 submanifold in X. We
denote the cut-and-paste (X − Y ) ∪ϕ Z by X(Y, ϕ, Z). In the case of Y = Z,
we denote such a surgery by X(Y, ϕ) and say it a twist. For simply-connectedclosed exotic 4-manifolds the following theorem is well-known.
Theorem 1.1 ([5],[11],[2]). For any simply-connected closed exotic 4-manifolds
X1 , X2 , there exist a contractible Stein manifold C, an embedding C ,→ X1 ,and a self-diﬀeomorphism t : ∂C → ∂C with t2 = id such that
X1 (C, t) = X2 .
This theorem says that a pair of a contractible Stein manifold and aself-diﬀeomorphism of the boundary causes “exoticity” of simply-connectedclosed smooth 4-manifolds. Studying corks is important for understanding
the exotic phenomenon of 4-manifolds. We give a definition of cork in a
generalized form.
Definition 1.2 (Cork). Let C be a contractible 4-manifold and t a selfdiﬀeomorphism ∂C → ∂C on the boundary. If t cannot extend to a map
C → C as a diﬀeomorphism, then (C, t) is called a cork.
For a pair of exotic two 4-manifolds X1 , X2 and a smooth embedding
C ,→ X1 , if X1 (C, t) = X2 , then (C, t) is called a cork for X1 , X2 . We callthe deformation X1 → X1 (C, t) a cork twist.
Date: February 12, 2016.
1991 Mathematics Subject Classification. 57R55, 57R65.
Key words and phrases. Stein manifold, finite order cork, exotic 4-manifold.
The author was partially supported by JSPS KAKENHI Grant Number 26800031.
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Note that this definition of cork are weaker than that of the usual one interms of the following two points. The definition here does not assume that
C is Stein and t satisfies t2 = id. If the 4-manifold C of a cork (C, t) is Stein,then (C, t) is called a Stein cork.
1.2. Order n cork. We define order n cork.
Definition 1.3 (Order n cork). Let (C, t) be a cork. If the following conditions are satisfied, then we call (C, t) an order n cork:(1) the composition t ◦ t ◦ · · · ◦ t = ti (0 < i < n) cannot extend to any
diﬀeomorphism C → C.
(2) tn can extend to a diﬀeomorphism C → C.
This number n is called the order of the cork (C, t).
Let X be a pair of n mutually exotic 4-manifolds {X = X0 , · · · , Xn−1 }.
If there exists an embedding C ,→ X such that Xi = X(C, ti ), then (C, t) iscalled a cork for this collection X.
The existence of finite order corks has been not known except for order2.
1.3. Aims. Let C be a contractible 4-manifold. One of our aims of thisarticle is to construct infinite families of order n corks for each n > 1.
Another aim is to give a technique to show that the map t for a twist (C, t)cannot extend to inside C as a diﬀeomorphism, i.e, (C, t) is a cork.
Freedman’s result [6] says that the diﬀeomorphism on ∂C extend to aself-homeomorphism on C.
Infinite order corks are not known so far.
1.4. Results. Let (C(m), τ (m)) be a pair defined as the handle diagram
as in Figure 1 and the diﬀeomorphism τ (m) is order 2. In the case of

Figure 1. Corks (C(m), τ (m)).
m = 1, (C(1), τ (1)) is the Akbulut cork in [4]. (C(m), τ (m)) is an example
of an order 2 Stein cork (Proposition 3.1). By positioning several copies of
the attaching spheres of this cork on the boundary of the 0-handle, we give
examples of finite corks. One of the main theorems is the following.
Theorem 1.4. Let n, m be integers with n > 1 and m > 0. There exists anC ). The handle decomposition and the map τ Corder n cork (Cn,m , τn,mn,m aredescribed in Figure 2. Furthermore, Cn,m is an order n Stein cork.

FINITE ORDER CORKS

Figure 2. The handle decomposition of Cn,m .

Figure 3. The handle decomposition of C3,m after an isotopy.
The number −m in any box stands for a −m full twist.
We will define other variations Dn,m and En,m in Section 2.5. Here we
give rough definitions of them. Dn,m is obtained by the exchange of all dots
and 0s of Cn,m . En,m is a 4-manifold modified as in Figure 5 of Cn,m . The
D andcase of n = 3 is described as in Figure 6. The diﬀeomorphisms τn,m
E are the rotations by angle 2π/n in the same way as τ C .
τn,mn,mThe reason why we treat these examples is to show the existence of manycorks without the direct aid of Stein structure. In other words, even when
we do not know whether Dn,m and En,m are Stein manifolds, our techniqueED
cannot extend to inside Dn,m and En,m asand τn,m
can show that τn,m
diﬀeomorphisms respectively. Here, as examples, we state the second and3

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Figure 4. The handle decomposition of Dn,m .

Figure 5. A modification of Cn,m into En,m .

Figure 6. Handle decomposition of E3,m and a diﬀeomorE .
phism τn,m

FINITE ORDER CORKS

third main theorems in the form including technical statements.
D )). Let n be an integer with n > 2.
Theorem 1.5 (Cork-ness of (Dn,m , τn,mThen there exists an embedding Dn,m ⊂ D2,m such that for the embedding,D ) such that thethere exists a diﬀeomorphism ψ : D2,m → D2,m (Dn,m , τn,m
D on the boundaries.
diﬀeomorphism induces τ2,m
D ) is an order n cork.
In particular, (Dn,m , τn,m

Note that the first statement does not mean the induced diﬀeomorphism∂D2,m → ∂D2,m can extend to inside a diﬀeomorphism. The induced map
is the restriction of a rotation of D2,m − Dn,m of ψ to the one component of
the boundary.
E )). For a positive integers n, m
Theorem 1.6 (Cork-ness of (En,m , τn,mthere exists a suﬃcient large integer l and an embedding En,m ,→ Vl,n :=E )i ) is
E(l)#nCP 2 such that for any 0 < i < n the twist Vl,n (En,m , (τn,mdiﬀeomorphic to (2l − 1)CP 2 #(10l + n − 1)CP 2 .
In particular, En,m is an order n cork.

The integer l in this theorem is independent of m. The point is that allE )i ) for an embedding E
the nontrivial twists (En,m , (τn,mn,m ,→ Vl,n produces
exotic structures. Whether there exists a finite order cork for the collectionof mutually exotic 4-manifolds is not known yet.
1.5. More exotic 4-manifolds. In Section 3.3, we give a 4-manifold Wn,mand embedding Cn,m ,→ Wn,m . The cork twist is a candidate of the collectionof mutually exotic 4-manifolds.
Proposition 1.7. Let i be an integer with 0 < i ≤ n − 1. There exists a
simply-connected non-spin 4-manifold Wn,m,i with b2 = b− = n(n − 1)/2 anda homology sphere boundary. The manifold Wn,m,i is obtained by an ordern cork twist of a Stein manifold Wn,m . Each Wn,m,i is the i times blow-ups′
of a 4-manifold Wn,m,i
and is exotic to Wn,m .
′We do not know whether Wn,m,i
is a minimal 4-manifold.

Remark 1.8. In the definition of order n cork in [13], we imposed thecondition that the order of t is n. Here we slightly change it to the weakercondition (2) in Definition 1.3.
This remark says that the order of t as a map is “not” necessary to be thesame as the order of cork. In the last section we will illustrate an example
having the diﬀerence.
Proposition 1.9. Let (F, κ) be a pair of a 4-manifold and diﬀeomorphism
as in Figure 21. The map κ on ∂F has order 2 as a cork, however it is anorder 4 as a diﬀeomorphism.

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1.6. An action on Heegaard Floer homology. S. Akbulut and C. Karakurt
in [1] showed that the twist map t of an order 2 Stein cork (C, t) consisting of a symmetric diagram consisting of a dotted 1-handle and a 0-framed
2-handle induces an action on the Heegaard Floer homology HF + (∂C) nonC )trivially as an involution. We show that an order n Stein cork (Cn,m , τn,malso induces an order n map on Heegaard Floer homology HF + (∂Cn,m ).
Theorem 1.10. Let n, m be positive integers with n > 1. The maps
C )i |i = 0, · · · , n − 1}, which is isomorphic to Z/nZ, act eﬀectively on{(τn,mthe Heegaard Floer homology HF + (∂Cn,m ).
An action of a group G on a set S is called eﬀective, if for any e ̸= g ∈ Gthere exists an element x ∈ S such that g · x ̸= x. From this theorem, we
immediately the following proposition.
Proposition 1.11. There exist n Stein filling contact structures on ∂Cn,msuch that they are homotopic and contactomorphic but non-isotopic eachother.
Acknowledgements
This topic was come upon by the author while speaking with Masatsuna
Tsuchiya on November in 2014. I thank him for arguing with me. I gave
a talk about finite order corks in the Handle friendship seminar in Tokyo
Institute of Technology and also thank the audience for giving the helpfulcomments. I also thank Kouichi Yasui and Tetsuya Abe for giving me usefulcomments in the earlier version.
2. A 4-manifold Xn,m (x) for a {∗, 0}-sequence.
2.1. Extendability of a composite boundary diﬀeomorphism. We
defined the order of cork in Section 1. The definition is slightly diﬀerent
from that defined in [13] as mentioned in Section 1.5. To note the welldefinedness of the order of cork, we show the following fundamental lemma.
Lemma 2.1. Let X be a manifold and φ, ψ boundary self-diﬀeomorphisms∂X → ∂X. If φ, ψ extend to inside X → X as a diﬀeomorphism, then sodoes φ ◦ ψ.
Proof. We attach the cylinder ∂X ×I to X by using φ : ∂X ×{1} → ∂X.
The extendability problem of (X, φ ◦ ψ) is equivalent to that for a pair(X ∪φ (∂X × I), ψ), where ψ induces a map ψ : ∂X × {1} → ∂X × {1}.
Since φ : ∂X → ∂X extends to inside, by an identification X → φ(X),X ∪φ (∂X × I) is diﬀeomorphic to X with the boundary point-wise fixed.
The problem is reduced to the extendability problem of ψ : ∂X → ∂X.
From the assumption, therefore, ψ ◦ φ extends to inside.
□
From this lemma if a self-diﬀeomorphism φ on the boundary of a manifold
extends to inside as a diﬀeomorphism, then so does φn .

FINITE ORDER CORKS

2.2. Constructions of Xn,m (x). Let m be a positive integer and n an
integer with n > 0. Let Ln,m denote 2n components link (located like awheel) as in the diagram in Figure 2. We denote the components by
Ln,m = {α0 , α1 , · · · , αn−1 , β0 , β1 , · · · , βn−1 },where the number −m in any box stands for a −m full twists. The ncomponents {αi |i = 0, 1, · · · , n − 1} (here called them radial components)of them lie in the radial direction about the center of the rotation. The rest
n-components {βi |i = 0, 1, · · · , n − 1} in Ln,m (here we call them circularcomponents) are put in the form that makes a circuit from a radial component to the same component. The linking number between αi and βi is one
and other linking numbers between αi and βj (i ̸= j) are all zero.
Let x = (x0 , x1 , · · · , xn−1 ) be a {∗, 0}-sequence with xi = ∗ or 0. If xi = ∗,then we describe a dot on the circle αi and 0 near the circle βi . If xi = 0,then we describe 0 near the circle αi and a dot on the circle βi . According
to this process, we get a framed link diagram. Xn,m (x) is defined to be a
4-manifold obtained by this framed link diagram.
From the construction, we have H∗ (Xn,m (x), Z) ∼= H∗ (B 4 , Z), where B 4
is the 4-ball. We note that X1,m (∗) = X1,m (0) = C(m) holds.
We can also construct Xn,m (x) in terms of branched cover of C(m). The
n-fold branched cover of C(m) along the slice disk of Kn,m as in Figure 7
is Xn,m (0, · · · , 0). Doing several times cork twists (C(m), τ (m)) for {αi , βi }with xi = ∗ in the sequence x, we get Xn,m (x).

Figure 7. A slice knot Kn,m on ∂C(m).
Lemma 2.2. Let x be any {∗, 0}-sequence. Xn,m (x) is a contractible 4manifold.

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Proof.
We show Xn,m (x) is simply-connected. For 0 ≤ i ≤ n − 1exchanging xi as ∗ → 0 corresponds to a cork twist of C(m) = 0-handle ∪{αi , βi }. Thus, this exchange does not change the fundamental group, i.e.,
π1 (Xn,m (· · · , ∗, · · · )) ∼= π1 (Xn,m (· · · , 0, · · · )).
We may show π1 (Xn,m (0, · · · , 0)) is trivial. Dotted circles in the diagramof Xn,m (0, · · · , 0) have a separated position with each dotted circles. Since
any αi does not link with βj with i ̸= j. The presentation of the fundamentalgroup is the same as π1 (♮nC(m)). Thus, this group is the trivial group. Thismeans π1 (Xn,m (x)) = e for any {∗, 0}-sequence.
Thus Xn,m (x) is a contractible 4-manifold.
□Definition 2.3 (Cn,m , Dn,m , and Fn,m ). We define Cn,m , Dn,m and Fn,m
to be
Cn,m = Xn,m (∗, 0, · · · , 0),
Dn,m = Xn,m (0, ∗, · · · , ∗),andFn,m = X2n,m (0, ∗, 0 ∗ · · · , 0, ∗).
See Figure 2 and 4 for Cn,m and Dn,m and see Figure 21 for F2,m .
We show the following proposition.
Proposition 2.4.
(1) C2,m = D2,m holds.
(2) Cn,m , Dn,m , En,m and Fn,m are contractible 4-manifolds.
Proof. (1) The handle decompositions of C2,m and D2,m are the same
C and τ D are the exchange of dots and 0s.
and the diﬀeomorphisms τ2,m2,m(2) Lemma 2.2 says that Cn,m , Dn,m and Fn,m are contractible. We canalso show that En,m is contractible in the same way as Lemma 2.2.
X . Moving each component of L
2.3. A diﬀeomorphism τn,mn,m asαi → αi−1and
βi → βi−1 ,X : ∂X
we get a diﬀeomorphism τn,mn,m (x) → ∂Xn,m (x). Here we consider
X when X = C, D
the suﬃces as elements in Z/nZ. The diﬀeomorphism τn,m
is a rotation by angle 2π/n as defined in Section 1.4.

2.4. An isotopy. We give an isotopy of Ln,m as described in Figure 8.
First, we move β0 to the innermost position. Next, we move β1 to thesecond innermost position. In the same way as above we move all βi . By
using the isotopy, we can get a handle diagram presentation of Xn,m (x). Asa diagram after isotopy, see Figure 3.

FINITE ORDER CORKS

Figure 8. An isotopy of a handle diagram of Cn,m .
2.5. Another variation En,m .
Definition 2.5. We define En,m to be the manifold obtained by the modification of Cn,m as in Figure 5.
The handle diagram for E3,m after the same isotopy as above is Figure 6.
In the same way as Cn,m or Dn,m each pair of a dotted 1-handle and 0framed 2-handle with linking number 1 in En,m consists of the cork C(m)(see Figure 9).
3. Proofs of main results
3.1. The cork-ness of Cn,m . Due to Gompf’s result [9] in order to see that
C(m) and Cn,m admit Stein structure, we may deform the handle diagramsof C(m) and Cn,m into Legendrian links with some Thurston-Bennequin condition. On the standard position of #nS 2 × S 1 in [9], which is the boundaryof the end sum ♮nD3 × S 1 , we put Legendrian links with all framings tb − 1,
where tb is the Thurston-Bennequin number of each Legendrian knot.
Here we show the following.

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Figure 9. The Akbulut cork C(1) embedded in En,1 .
Proposition 3.1. (C(m), τ (m)) is an order 2 Stein cork.
Proof. The Stein structure on C(m) is due to Figure 10. Here, thebox with number −m + 1 stands for a (−m + 1)-full twist with a Legendrian
position as the second equation in Figure 11.
□

Figure 10. Stein structure on C(m).

Figure 11. Legendrian positions of strings with negative twists.
We prove Theorem 1.4.
Proof of Theorem 1.4. We deform the handle diagram in Figure 3 as in

FINITE ORDER CORKS

Figure 12. The last picture in Figure 12 can be easily changed by isotopy
into a Legendrian link in the standard position of a connected sum of several
copies of S 2 ×S 1 as in Figure 13. For the case of n = 4, the handle diagram
is Figure 14.
C )i cannot extend
We show that for any integer i with 1 ≤ i ≤ n − 1, (τn,m
to inside Cn,m . One attaches a −1-framed 2-handle h to the meridian of
βn−i in Cn,m . The resulting 4-manifold
Zn−i = Cn,m ∪βn−i his a Stein 4-manifold. However the attaching sphere of the correspondingC )i ) is a meridian of the 0-framed 2-handle. Thus
βn−i in Zn−i (Cn,m , (τn,mwe can construct an embedded sphere with self-intersection number −1 bytaking the union of the core disk of h and compressing disk of the meridianin ∂Cn,m . On the other hand, in any Stein 4-manifold, there never existC )i )
any embedded −1-sphere, for example see [3]. Thus Zn−i (Cn,m , (τn,mC )i ) are exnever admit any Stein structure, hence Zn−i and Zn−i (Cn,m , (τn,mCiotic 4-manifolds. This implies (τn,m ) cannot extend to inside Cn,m . Since
C )n = id, clearly (τ C )n can extend to a diﬀeomorphism on C
(τn,m
n,m .
n,m
C ) is an order n cork.
Therefore (Cn,m , τn,m□
The following Lemma 3.2 and Corollary 3.5 are key lemma and corollary
for the results on cork in this article.
Lemma 3.2. Let x be a {∗, 0}-sequence with x ̸= (∗, ∗, · · · , ∗), (0, 0, · · · , 0).
X ) is a cork twist.
Then the twist (Xn,m (x), τn,m
Proof. By permuting x as (x0 , x1 , · · · , xn−1 ) 7→ (x1 , · · · , xn−1 , x0 ) in
several times, we may assume that the sequence is x = (∗, 0, x2 , · · · , xn−1 ).
For 2 ≤ i ≤ n − 1 if xi = ∗, then attaching a 0-framed 2-handle to themeridian of αi in Xn,m (x) and canceling with αi , we can make a separated
0-framed 2-handle in the diagram of Xn,m (x) ∪ 2-handle. Canceling the 0framed 2-handle with a 3-handle, we get Xn,m (x) ∪ 2-handle ∪ 3-handle =Xn−1,m (x′ ), where x′ is (∗, 0, x2 , · · · , x̂i , · · · , xn−1 ). The hat means deletingof the component. This handle attachment means Xn,m (x) ⊂ Xn−1,m (x′ ).
The cobordism between ∂Xn,m (x) and ∂Xn−1,m (x′ ) is a homology cobordism. Iterating this process, we have Xn,m (x) ⊂ Xn′ ,m (∗, 0, 0, · · · , 0).
For 2 ≤ i ≤ n′ − 1 if xi = 0, then attaching a 2-handle to the meridian of
αi in Xn′ ,m (∗, 0, · · · , 0), we can move αi to the position in the first picture in
Figure 15. By doing the handle slide along the arrow in the picture, we get
the second picture. By sliding some components to the meridional 0-framed
2-handle, the components {αi , βi } with the 0-framed 2-handle is separated
as in Figure 15. Canceling the 0-framed 2-handle with a 3-handle, we getXn′ −1,m (∗, 0, · · · , 0). Iterating such a canceling, we get X2,m (∗, 0). Thus,
we get an embedding
Xn,m (x) ⊂ X2,m (∗, 0) ⊂ X1,m (∗) = C(m).
X
X be the 2π/n rotation as in Figure 2. The twist C(m)(XLet τn,mn,m (x), τn,m )is diﬀeomorphic to C(m), and by the twist, the diﬀeomorphism on the

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Stein structure

Figure 12. The isotopy of C3,1 to the Stein structure.

X extends to a selfboundary is mapped in the same way as τ (m). If τn,m
diﬀeomorphism on inside Xn,m , then τ (m) can extend to inside C(m). ThisX ) is a cork.
is a contradiction to Proposition 3.1. Therefore, (Xn,m (x), τn,m□
Note that these cobordisms give homology cobordisms.

Definition 3.3 (Shifting). Let x be a {∗, 0}-sequence. If Si is a cyclic map
acting on {∗, 0}-sequences as x = (x0 , x1 , · · · , xn−1 ) 7→ (x−i , x−i+1 , · · · , x−i−1 ),FINITE ORDER CORKS

Figure 13. A Stein structure of Cn,m .

Figure 14. A Stein structure of C4,m .

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Figure 15. Xn,m (∗, 0, · · · , 0) with 0-framed 2-handle.
we call Si a shifting map on the sequence. Here we consider the suﬃces as
Z/nZ.
Definition 3.4 (Period). Let x = (x0 , x1 , · · · , xn−1 ) be a {∗, 0}-sequence.
We callmin{p|Sp (x) = x, p > 0}
the period of x.
From the definition, the period is a divisor of n.
Corollary 3.5. Let x be a {∗, 0}-sequence with period N > 1.
X ) is an order N cork.
(Xn,m (x), τn,m

Then

Proof. For any integer 0 < i < N , there exists j such that Si (xj ) ̸= xj−i .
If there does not exist such j, then the period of x is less than or equal toi. This is contradiction. Canceling all handles but αj−i and βj−i as in the
proof in Lemma 3.2, we have an embedding:Xn,m (x) ⊂ X1,m (xj−i ) = C(1).
X )i exchanges the dotted
By the shifting map Si the self-diﬀeomorphism (τn,mX
is the 2π/n1-handle and 0-framed 2-handle on {αj−i , βj−i }, where τn,mrotation, since Si (xj ) ̸= xj−i .
X )i ) for the embedding gives the eﬀect (C(1), τ (m)).
This twist C(1)(Xn,m , (τn,mX )i ) cannot
By the same argument as the proof of Lemma 3.2, (Xn,m (x), (τn,mextend to inside Xn,m (x) as a diﬀeomorphism.
X )N
Since shifting map SN does not change x, the diﬀeomorphism (τn,mcan extend to Xn,m (x). As a result, (Xn,m (x), τn,m ) is an order N cork. □FINITE ORDER CORKS

3.2. Cork twist for embeddings relationship in {Cn,m }. Let k, n, m
be positive integers. There exist embeddings Cn,m ⊂ Cn+k,m , and Cn+k,m ⊂Cn,m according to Lemma 3.2.
Corollary 3.6 (Cork twist of Cn,m ). Let n be a positive integer. The cork
C ) for the first embedding above gives a diﬀeomorphism
twist Cn,m (C2,m , τ2,m
C ) and the boundary restriction φ| : ∂C
φ : Cn,m → Cn,m (C2,m , τ2,m
n,m →
∂
C
∂Cn,m coincides with τn,m .
Proof.
This assertion follows immediately from Lemma 3.2 and theargument below.
Suppose that n > 2. The inserting map of {∗, 0}-sequences(∗, 0) = (x0 , x1 ) 7→ (x0 , y, · · · , y, x1 ),where y = 0 gives an embedding C2,m ,→ Cn,m . Thus, the cork twist
Cn,m (C2,m , τ2,m ) give the eﬀect(x0 , y, · · · , y, x1 ) 7→ (x1 , y, · · · , y, x0 ).
C ).
This deformation corresponds to the cork twist (Cn,m , τn,m
Suppose that n = 1. The deleting map of {∗, 0}-sequences(∗) 7→ (0).
gives an embedding C2,m ,→ C1,m . Thus, the cork twist C1,m (C2,m , τ2,m )give the eﬀect (∗, 0) 7→ (∗) This deformation corresponds to the cork twist(C(m), τ (m)).
Suppose that n = 2. By taking the identity map C2,m → C2,m , thestatement is trivial.
□

Figure 16. An embedding C2,m ,→ C(m).

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Proposition 3.7. The Alexander polynomial of Kn,m is computed as follows:∆Kn,m (t) = 2tn − 5 + 2t−n .
Proof.
The Alexander polynomial of Kn,m is obtained by the Fox
calculus of the fundamental group of the knot exterior. The −m twist in
the diagram in Kn,m ⊂ ∂Cn,m is indiﬀerent from the fundamental groupof the exterior of the knot. Thus, the fundamental group is computed by
π1 (∂Cn,m − Kn,m ) ∼= π1 (∂Cn,0 − Kn,0 ). ∂Cn,0 is homeomorphic to S 3 .
Remark 3.8. For En,m , one can find the similar embeddings to those ofCn,m . Namely, there exist En,m ⊂ En+k,m and En,m ⊂ En+k,m such thatthese embeddings satisfy the same condition as that of Proposition 3.6
Proof of Theorem 1.5. By using the deleting map (0, ∗, · · · , ∗) 7→ (0, ∗)in Lemma 3.2, the current theorem is satisfied. Therefore, as a corollary ofLemma 3.2 and Corollary 3.5 we deduce (Dn,m , τn,m ) is an order n cork. □Proof of Theorem 1.6. Attaching 2-handles as the first diagram in
Figure 17, we obtain the last diagram in the figure. The 4-manifolds for thisdiagram can be embedded in E(l)#nCP 2 . This can be seen due to Figure 9.4
2
in [10]. Actually, if l ≥ ⌈ 2n+1
3 ⌉, then this embedding into E(l)#nCP canbe constructed.
We show that the twist by an embedding En,m ,→ Vl,n produces (2l −1)CP 2 #(10l + n − 1)CP 2 by using the argument in Exercise 9.3.4 in [10].
E )i ) contains the handle diagram as in Figure 18.
En,m ,→ Vl,n (En,m , (τn,m
By doing the handle slide as indicated by the arrow in the right hand sidein Figure 18, we get a CP 2 connected-sum component. By using a CP 2
formula (Figure 19), we get the third diagram in Figure 18. A separated
CP 2 component in the third diagram can be moved to the position before
separating by using the converse of the handle moves from the second picture
to the third picture in Figure 18. For the last figure in Figure 18 we canuse the method as Exercise 9.3.4. in [10]. As a result, for 0 < i ≤ n − 1 we
E )i ) ∼ (2l − 1)CP #(10l + n − 1)CP 2 ∼have a diﬀeomorphism Vl,n (En,m , (τn,m̸==EVn,l . This means that (En,m , τn,m ) is an order n cork.
□
Remark 3.9. Proposition 3.6 and Theorem 1.5 imply that even when C(m)cannot be embedded in a 4-manifold X, if the diagram of C(m) is containedin X as a sub-diagram and Cn,m ,→ X or Dn,m ,→ X with respect to the subC ) or (Ddiagram, then by doing cork twist (Cn,m , τn,mn,m , τn,m ) respectively,
exchanging the dot and 0 in the sub-diagram of X can be realized by an cork
twist Cn,m and Dn,m .
3.3. Exotic 4-manifolds Wn,m,i . If there does not exist any smoothly embedded −1-sphere in a 4-manifold, then we say the 4-manifold is minimal.
Proof of Proposition 1.7. Let Wn,m be a 4-manifold Cn,m with n(n −1)/2 2-handles attached. For 1 ≤ i ≤ n − 1, the attaching spheres are the iparallel meridians of βi in the diagram of Cn,m in Figure 2. The framings
are all −1. The diagram of W3,m is Figure 22. The manifold Wn,m isFINITE ORDER CORKS

Figure 17. An embedding En,m ,→ E(l)#nCP 2 .
simply-connected and b2 = n(n − 1)/2 and intersection form is the b2 direct
sum of ⟨−1⟩. Clearly W3,m is a Stein manifold and in particular minimal.
C )i ) for W
Performing the cork twist (Cn,m , (τn,mn,m , we get a 4-manifold
CWn,m,i = Wn,m (Cn,m , (τn,m)i ).

By this cork twist, the i parallel meridians are moved to the parallel meridians of β0 , which is a 0-framed 2-handle. Thus the i meridians can be′blow-downed. Hence, we get Wn,m,i = Wn,m,i# i CP 2 . Thus Wn,m,i andWn,m are exotic, because Wn,m is minimal.
□
The problem of whether {Wn,m,i }i=0,··· ,n−1 are mutually exotic 4-manifolds′
is remaining. If all Wn,m,ifor any i are minimal, then Wn,m,i are mutually
C ) is a cork for the collection {Wexotic 4-manifolds, i.e., (Cn,m , τn,mn,m,i }.
Proof of Proposition 1.9. Let F = F2,m be a contractible 4-manifold
F be a rotation by π/2. Since (∗, 0, ∗, 0)
defined in Definition 2.3. Let κ = τ2,m
is a period 2 {∗, 0}-sequence, (F, κ) is order 2 cork by Lemma 3.5.
□
4. A non-trivial action on HF + (∂Cn,m ).
In this section, as an application of finite order corks we prove Theorem 1.10. This theorem is a generalization of the main theorem in [1]. The

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MOTOO TANGE

E )i ).
Figure 18. A submanifold E(l)#nCP 2 (En,m , (τn,mFigure 19. A separated CP 2 formula.
terms used here are the same ones as those in [1]. The argument is parallelto Theorem 4.1 in [1].
C )i induces
Proof of Theorem 1.10. We may show that the map (τn,mi+Ca non-trivial action on HF (∂Cn,m ). Let τi denote (τn,m ) . Let ξ be thecontact structure on ∂Cn,m induced from the Stein structure on Cn,m . As
described as in Figure 2 in [1], we attach a 2-handle H along a trefoil linked
with βn−i with framing 1. We denote Ui = Cn,m ∪βn−i H. This manifold is

FINITE ORDER CORKS

Figure 20. The handle decomposition of W3,m .

Figure 21. A contractible 4-manifold F2,m and a diﬀeomorF with order 4 as a diﬀeomorphism.
phism τ2,ma Stein manifold because the maximum Thurston-Bennequin number of thetrefoil is 1 and the Stein structure of Cn,m extends to H.

C )i .
Figure 22. The twist of Ui via (τn,mLet V be a concave extension of (∂Ci , ξ) of H due to [7]. Thus, Xi =Cn,m ∪ V is a closed symplectic structure with b+
2 > 1. We define the twist
via τi by Xi′ = Cn,m ∪τi V . In this manifold, we can find an embedded
self-intersection number 1 torus. Here we use the following theorem. Here
ΦX,s is the Ozsváth-Szabó 4-manifold invariant.

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Theorem 4.1 (Ozsváth-Szabó [8]). Let X be a closed 4-manifold. Let Σ ⊂X be a homologically non-trivial embedded surface with genus g ≥ 1 andwith non-negative self-intersection number. Then for each Spinc structures ∈ Spinc (X) for which ΦX,s ̸= 0, we have that|⟨c1 (s), [Σ]⟩| + [Σ] · [Σ] ≤ 2g − 2
The following is another version of the adjunction inequality along with a
non-vanishing result of the 4-manifold invariant for Lefschetz fibrations.
Hence, the Ozsváth-Szabó 4-manifold invariant for X ′ has no basic class.
Here we obtain the same computation as in [1]:
−
+
mix
mix
FC+n,m ,s0 (c+ (ξ)) = ±FC+n,m ,s0 ◦ FV,s(Θ−(−2) ) = FX,s (Θ(−2) ) = ±Θ(0) ,and on the other hand,
−
mix
mix
FC+n,m ,s0 (τi∗ (c+ (ξ))) = FC+n,m ,s0 ◦ τi∗ ◦ FV,s(Θ−(−2) ) = ±FX ′ ,s′ (Θ(−2) ) = 0,

where s is the canonical spinc structure on symplectic manifold X and s′
is structure, the induced spinc structure on X ′ , and s0 is the restrictionof s and s′ to Cn,m . Here c+ (ξ) is Ozsváth-Szabó’s contact invariant for
C , ξ). We use the equality F mix (Θ− ) = ±c+ (ξ) in [12], where ξ is a(∂Cn,mW,ξ
(−2)
contact structure on ∂W with torsion c1 .
These two inequalities above means that c+ (ξ) and τi∗ (c+ (ξ)) are distinct
C act on HF + (∂C∗∗elements. Thus, τn,mn,m ) eﬀectively. Namely, ξ, τ1 (ξ), · · · , τn−1 (ξ)
are distinct elements.
□
Proof of Proposition 1.11. Let ξ be a contact structure on ∂Cn,m induced by the Stein structure. Let ξi denote τi∗ (ξ). Each two structures ξi , ξjare homotopic 2-plane fields. For, there exist a trivial cobordism between
these contact 3-manifolds, hence ξi , ξj are the same 3-dimensional invariantsby [9]. This means that these contact structures are homotopic each other.
The diﬀeomorphism τi−j gives a contactomorphism from ξi to ξj . However,ξi , ξj are not isotopic because c+ (ξi ) and c+ (ξj ) are distinct. Furthermore,ξi is Stein filling.
□
5. Problems
Here we raise the several problems.
C ) is a cork for the collection {W
Problem 5.1. Show that (Cn,m , τn,mn,m,i }i=0,1,··· ,n−1 .

Problem 5.2. Show that Dn,m , En,m are also finite order Stein corks. In
general for any {∗, 0}-sequence x, show that Xn,m (x) is a Stein manifold.
References[1] S. Akbulut and C. Karakurt, Action of the cork twist on Floer homology, Proceedingsof the Gökova Geometry-Topology Conference 2011, 42-52, Int. Press, Somerville,
MA, 2012[2] S. Akbulut and R. Matveyev, A convex decomposition theorem for 4-manifolds, Internat. Math. Res. Notices 1998, no. 7, 371-381.
[3] S. Akbulut and R. Matveyev, Exotic structures and adjunction inequality, Turkish J.
Math.21(1997)47-53

FINITE ORDER CORKS[4] S. Akbulut, A fake compact contractible 4-manifold, J. Diﬀ. Geom. 33(1991)[5] C. L. Curtis; M. H. Freedman; W. C.Hsiang; R. Stong, A decomposition theorem forh-cobordant smooth simply-connected compact 4-manifolds, Invent. Math. 123 (1996),
no. 2, 343-348.
[6] M. Freedman, The topology of four-dimensional manifolds, J. Diﬀerential Geom. Volume 17, Number 3 (1982), 357-453.
[7] P. Lisca and Matic, Tight contact structures and Seiberg-Witten invariants, Invent.
Math. 129(1997)509-525[8] Ozsváth and Szabó, Holomorphic triangles and invariants for smooth four-manifolds
Adv. Math. 202 (2006), no. 2, 326-400[9] R. Gompf, Handlebody construction of Stein surfaces. , Ann. of Math. (2) 148 (1998),
no. 2, 619-693.
[10] R. Gompf and A. Stipsicz, 4-manifolds and Kirby calculus, Graduate Studies in Mathematics, 20. American Mathematical Society, Providence, RI, 1999.
[11] R. Matveyev, A decomposition of smooth simply-connected h-cobordant 4-manifolds,
J. Diﬀerential Geom. 44 (1996), no. 3, 571-582.
[12] , O. Plamenevskaya, Contact structures with distinct Heegaard Floer invariants ,Math. Res. Let. 11,(2004) 547-561.
[13] M. Tange, A plug with infinite order and some exotic structures, to appear to
JGGT(2015)
Institute of Mathematics, University of Tsukuba, 1-1-1 Tennodai, Tsukuba,
Ibaraki 305-8571, Japan
E-mail address: tange@math.tsukuba.ac.jp

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