NON-EXISTENCE THEOREMS ON INFINITE ORDER
CORKS
MOTOO TANGE

Abstract. Suppose that X, X ′ are simply connected closed exotic 4manifolds. It is well-known that X ′ is obtained by an order 2 cork twistof X. We show that in the case of infinite order cork, this existence
theorem does not always hold.

1. Introduction
1.1. A fact for cork twist. In smooth 4-manifolds the following existence
theorem of a cork is well-known.
Fact 1.1 ([9],[4]). Let X, X ′ be simply-connected closed exotic smooth 4manifolds. Then there exists a contractible 4-manifold C in X such thatX ′ = (X − C) ∪τ C and τ 2 = id.
Furthermore, as such a manifold C we can take a Stein manifold [1]. ‘Exotic’ means that manifolds are homeomorphic but non-diﬀeomorphic eachother. The manifold obtained by removing a submanifold Y ⊂ X with embedding i and regluing Y via τ is denoted by X(i, Y, τ ). We may omit theembedding map i in the notation, if the map is understood in that context.
Here we call such a surgery simply twist. Hence, cork means a localizationof exotic structure.
1.2. Motivation and results. As Fact 1.1 mentioning, any exotic two 4manifolds X and X ′ have an involutive relationship with respect to a cork
twist. What we issue is the point of whether the existence holds for an
infinite family. In this paper we give a negative answer (Main theorem 1) for
this question. In the local situation we have a natural question of whethera (generalized) cork twist is a result of an inner cork or not. We give anegative answer (Main theorem 2) for this question as well.
1.3. Finite, infinite order cork, and Main theorem 1. Let (C, τ ) bea pair of a smooth manifold C and a boundary diﬀeomorphism τ : ∂C →∂C. If τ extends to a homeomorphism on C but cannot extend to any
diﬀeomorphism on C, then τ is called non-trivial (otherwise trivial). If C
Date: September 26, 2016.
1991 Mathematics Subject Classification. 57R55, 57R65.
Key words and phrases. Infinite order cork.
The author was partially supported by JSPS KAKENHI Grant Number 26800031.
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MOTOO TANGE

is a contractible and τ is non-trivial, then the pair (C, τ ) is called a cork.
Freedman’s result [6] says that C is a contractible and τ cannot extend
to any diﬀeomorphism on C, then (C, τ ) is a cork. When we replace thecontractible condition with the non-contractible one, we call (C, τ ) a noncontractible cork. Then the map τ is called cork map or non-contractible
cork map. The order of a cork (or non-contractible cork) is the minimal
positive number of n that τ n can extend to a diﬀeomorphism on C.
The author in [14] illustrates an example of a non-contractible cork. Recently, by the author [13] and Auckly, Kim, Melvin, and Ruberman [2] finite
order corks are found. Right after the discoveries, Gompf in [7] found infinite
order corks.
Theorem 1.2 ([7]). Suppose that Kn is the 2n-twist knot. Then there exists
an infinite order cork (C, f ) satisfying XKn = X(C, f n ).
Here the 4-manifold X need have 2 vanishing cycles isotopic to the meridian of the knot-surgery. At the point that (C, f ) produces Fintushel-Stern’sknot-surgeries, this cork is very exciting object.
We prove the following non-existence theorem on infinite order cork. Here
we denote by F the order 2 field Z/2Z.
Main theorem 1. Suppose that {Xn } is a Z-family of exotic oriented closed
4-manifolds with b+
2 > 1 giving infinite OS-invariants with F-coeﬃcient.
Then, there exists no infinite order cork (C, τ ) such that {Xn } = {X(C, τ n )}.
This theorem would be true even if one replaces OS-invariant with SeibergWitten invariant with F-coeﬃcient, because of the equivalence of the OSinvariant and the Seiberg-Witten invariant. This equivalence for 4-manifolds
is still open now.
For a closed spinc 4-manifold (X, s) the OS-invariant ΦX,s is a smooth
4-manifold invariant
ΦX,s ∈ F.
Then we have a polynomial∑

ΦX,s · eP D[c1 (s)] =: ΦX .

s∈Spinc (X)We call the polynomial OS-invariant. As an application, the following corollary holds.
Corollary 1.3. Suppose that Tn is the (2, 2n + 1)-torus knot. Then for any
integer m with m ≥ 2 the family {E(m)Tn } cannot be constructed by twisting
an infinite order cork.
Compared this corollary with Theorem 1.2, we know that the two situations are contrasting. The F-reductions of {∆Tn (t)} are infinite, i.e.,
{ n
}
∑
k
#
t |n ∈ Z = ∞,k=1

NON-EXISTENCE THEOREMS ON INFINITE ORDER CORKS

while the F-reductions of {∆Kn (t)} are finite, precisely saying#{1, t − 1 + t−1 } = 2.
Here we denote by ∆K (t) the Alexander polynomial of K. Depending on theknot, the existence of infinite order cork for Fintushel-Stern’s knot-surgery
changes.
This theorem means that the OS-invariants with F-coeﬃcient of 4-manifolds
obtained from a single (finite or infinite order) cork are finite variations.
Thus, immediately, we have the following corollary:
Corollary 1.4. Let C be a contractible 4-manifold. Let r be a rank of
HF − (∂C, F)/(U = 0). If C admits a G-cork with a G-eﬀective embedding∏rkwith distinct F-coeﬃcient OS-invariants, then |G| ≤ r−1
k=0 (2 − 2 ) holds.
Question 1.5. Let {Xn } be an exotic family of 4-manifolds (e.g., with
distinct Z-coeﬃcients OS-invariants). Suppose that {Xn } have finite Fcoeﬃcients OS-invariants. Then does there exist an infinite order cork (C, τ )which produces {Xn }?
A remaining question is a characterization of the finite family which is
produced by a finite order cork (C, τ ).
Question 1.6. Let {Xk |k = 0, · · · , n − 1} be a finite family of exotic 4manifolds. When does there exist an order n cork (C, τ ) such that the family
is obtained by cork twists of (C, τ ).
Let C be a 4-manifold and D a contractible submanifold of C with dim C =dim D and with ∂D smoothly embedded in the interior of C. Let i be theidentity map ∂C → ∂C. Thus, (C, i) is a trivial twist. Let g be a boundary
diﬀeomorphism of D. Suppose that there exists a diﬀeomorphism F fromthe twist C(D, g) to C. Then i induces a diﬀeomorphism ∂C → ∂C(D, g). We
define the composition F −1 |∂C ◦ i by j. We call (C, j) (or (D, g)) an induced
twist of (D, g) (or core twist of (C, f ) respectively).
F

C −−−−→ C(D, g)xinclusioni∂C −−−−→ ∂C(D, g)
It is already not clear whether (C, j) is trivial. Then we denote it by(D, g) ⊂ (C, f ).
Definition 1.7 (Core cork and induced cork.). Suppose that (D, g) ⊂ (C, f ).
If (C, f ) is a cork, then the twist (D, g) is also a cork. In this case we call(D, g) a core cork of (C, f ).
For the case where (C, f ) or (D, g) is a plug or non-contractible cork we use
the same terminology ⊂ in the similar situation. Even if a cork twist (D, g)
induces a twist (C, f ), then (C, f ) is not always a cork (or non-contractiblecork) twist. The orders of (C, f ) and (D, g) do not always agree with each3

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MOTOO TANGE

D )other. For example, in [14] the author proved cork-ness of (Dn,m , τn,mD ) ⊂ (C(m), τ (m)) and what(order n) by using an induced twist (Dn,m , τn,m
(C(m), τ (m)) is an order 2 Stein cork.
Furthermore, we consider the following concept for a family version ofcore (and indued) cork.

Definition 1.8 (Core G-cork, induced H-cork). Let (D, G) be a G-cork
and a submanifold in a 4-manifold C with boundary and ∂D ⊂ C smoothly
embedding. Assume that C − D is not diﬀeomorphic to a cylinder of ∂C.
If any g ∈ G gives an induced twist (D, g) ⊂ (C, h) and the correspondence
g 7→ h produces an isomorphism∼=

G → H ⊂ Diﬀ(∂C)
into a subgroup H, then (D, G) is called a core G-cork of (C, H) and (C, H)
is called an induced H-cork of (D, G). Then we denote it by(D, G) ⊂ (C, H).
In [13], we prove the Z2 -cork (C(1), {τ (1)}) contains a core Z2 -cork
C(C2,1 , {τ2,1}) ⊂ (C(1), {τ (1)}),

because, ∂C2,1 and ∂C(1) are not diﬀeomorphic homology spheres becauseof SnapPea computation. Therefore, C(1) − C2,1 is not diﬀeomorphic to thecylinder. It is an open question whether ∂C2,m ̸∼= ∂C(m) or not for any
m. The motivation of core cork is to replace a cork or non-contractible cork
twist with a new (or possible ‘universal’) reasonable cork.
To find a cork in a wider situation we would like to search a cork in a
non-contractible cork. As an application of Main theorem 1 we show thefollowing theorem.
Main theorem 2. There exists a non-contractible Z-cork (P, Z) such that
(P, Z) never contain any core Z-cork.
We give a natural question:
Question 1.9. Let H be a finite group. Does any non-contractible H-cork
contain a core G-cork D with (D, G) ⊂ (C, H)?
1.4. A Stein plug (Q, ϕ) with b2 (Q) = 1 changing any crossing of
Fintushel-Stern’s knot-surgery. Let (P, φ) be a plug with b2 = 2 which
is defined in [14]. The last assertion in this paper is that it is not a plug withthe minimal b2 which gives rise to any crossing change of Fintushel-Stern’sknot-surgery. Let Q be a 4-manifold obtained by attaching a 2-handle along
52 with 0-framing. The 4-manifold Q is a submanifold in P naturally. See
the first handle diagram of Q in Figure 4. Hence, we have ∂Q ∼= S03 (52 ).
Then we prove the following:
Proposition 1.10. There exists a diﬀeomorphism ϕ : ∂Q → ∂Q such that(Q, ϕ) is a Stein core Z-plug of (P, φ).

NON-EXISTENCE THEOREMS ON INFINITE ORDER CORKS

Figure 1. A Stein structure on Q.
A Stein structure on Q is presented in Figure 1. This Z-plug (Q, ϕ) produces infinitely many exotic Fintushel-Stern’s knot-surgeries. This meansthat the action on the Heegaard Floer homology should admit infinite order.
Let sk be a spinc structure with ⟨c1 (sk ), h⟩ = 2k, where h is a generatorin H2 (∂Q).
The Heegaard Floer homology of ∂Q is as follows:{ −−T(− 5 ) ⊕ T(−
k=0
7 ⊕ F(− 5 )
−
)
∼
2
2
2
HF (∂Q, sk ) =F[U ]/(U k − 1)k ̸= 0.
This computation will be done in Section 3.2. The action on the Heegaard Floer homology of ∂Q should be eﬀective. This fact is contrast to
Main Theorem 1. To investigate the mechanism that any crossing change ofFintushel-Stern’s knot surgery changes the diﬀerential structures in terms of
Heegaard Floer homology might become a help to understand exotic structures on 4-manifolds.
Acknowledgements
The main results were produced by an argument with Robert Gompf
who gave me suggestions. The author thanks him for his helps for thisearlier manuscript of this paper. Furthermore, the author is deeply grateful
with arguments with Kouichi Yasui in the Hiroshima University TopologyGeometry seminar.
2. Preliminaries and proofs of Main theorem 1 and 2
2.1. Knot-surgery. Let K be a knot in S 3 . Let X be a 4-manifold with asquare zero embedded torus T . Then the performance
XK = [X − ν(T )] ∪ [(S 3 − ν(K)) × S 1 ]is a (Fintushel-Stern’s) knot-surgery along K. The gluing map is indicatedin [5]. The notation ν(·) stands for an open neighborhood of a submanifold.
T. Mark in [8] proved the knot-surgery formula of (F-coeﬃcient) OzsváthSzabó’s 4-manifold invariant.
(1)

ΦXK = ΦX · ∆K (t).

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MOTOO TANGE

This is an Ozsváth-Szabó’s invariant counterpart of the Seiberg-Witten
formula of Fintushel-Stern’s knot-surgery in [5]. Here we have to notice
Sunukjian’s result [12] that the Alexander polynomial distinguishes smooth
structures of Fintushel-Stern’s knot surgeries.
2.2. Proof of Main theorem 1. Suppose that {Xn } is an exotic Z-familyof closed 4-manifolds with b+ (X) > 1 having infinite OS-invariants with Fcoeﬃcient and these are produced by cork twists by an infinite order cork(C, τ ). If Xn are not closed, then the same argument works by the relative
OS-invariant.
By permuting the order of Xn , we have X(C, τ n ) = Xn . The group⟨τ ⟩ ∼= Z acts on ∂C and the action induces a homomorphism on HF − (∂C).
The induced isomorphism τ∗ on HF − (∂C) keeps the absolute grading.
The grading shift of the action is calculated from the Euler number and thesignature of the cylinder I × ∂C. These invariants of the cylinder are all
zero. HFd− (∂C) with a fixed grading d is a finite abelian group which isisomorphic to F for suﬃcient small d’s. Hence there exists a positive integerm such that τ∗m is the identity.
Here we consider Xn as the gluing of three 4-manifolds {C, M, Ṽ }, where
Xn = [C ∪τ n M ] ∪N Ṽ and M is a cobordism ∂C → ∂ Ṽ . Furthermore,
we assume that N is an admissible cutting of Xn (defined in [11]). The
existences of these cuttings N are guaranteed in [11]. Deleting two 4-ballsin the interior in Xn , we give the composition Wn of three cobordisms:
C

MV

Wn : S 3 →0 ∂C → N → S 3 ,where actually the cobordism Wn is twisted on ∂C by action τ n and V is Ṽ
with a 4-ball deleted. Here the mixed invariant on Wn becomes as follows:
+
−
mix
FW
= FV,s◦ FM,s◦ τ∗n ◦ FC−0 ,s1 : HF − (S 3 ) → HF + (S 3 ),
n ,s0
3
2

where s|Wn = s0 , s|C0 = s1 , s0 |M = s2 and s0 |V = s3 . Recall the OSmix (U d · Θ− ) = Φ+invariant ΦXn ,s ∈ F is defined by FW
Xn ,s · Θ , wheren ,sd = (c21 (s)−2χ(Xn )−3σ(Xn ))/4. Since {τ∗n |n ∈ Z} has a finite variation, themix
mixed invariant FW
is also finite variations with respect to n. Thus then ,s0csets {ΦXn ,s |s ∈ Spin (Xn )} are also finite variations only with respect to n.
This contradicts that {Xn } has infinite OS-invariants with F-coeﬃcient. □By a corollary we have the following:
Corollary 2.1. Regardless of the order of the cork, the variations of Fcoeﬃcient OS-invariants by a single cork are at most finite.
In the case of the Z-coeﬃcient invariant, the variations are not always
finite as a Gompf’s example in [7] implies.
Proof of Corollary 1.3. Let Tn be the (2, 2n + 1)-torus knot. Due tothe OS-invariant formula (1) of E(m)Tn with the F-coeﬃcient, we haveΦE(m)Tn = (t − t−1 )m−2 (tn − tn−1 + · · · − t−n+1 + t−n ) mod 2.
These give infinite OS-invariants. From Main theorem 1, the family {E(m)Tn |n ∈Z} never be produced by cork twists of an infinite order cork.
□

NON-EXISTENCE THEOREMS ON INFINITE ORDER CORKS

This proof means that for a family {Kn } of knots, if E(m)Kn is constructedby an infinite order cork, then #{∆Kn mod 2} < ∞. In fact 2n-twist knot
Kn in [7] is {∆Kn (t) mod 2} = {1, t − 1 + t−1 }.
Proof of Corollary 1.4. If a G-cork twist gives distinct F-coeﬃcient
OS-invariants, then the action is eﬀective on HF − (∂C, F)/(U = 0) = Fr .
We note that the action is U -equivariant. The induced action become aninvertible linear action on Fr . Hence, we obtain G ⊂ GL(r, F). Then we∏rkhave |G| ≤ |GL(r, F)| = r−12
k=0 (2 − 2 ).
2.3. Proof of Main theorem 2. Let (P, φ) be the plug defined in [14].
Namely, P and φ are described in Figure 2 and 3 respectively.

Figure 2. P .

Figure 3. The diﬀeomorphism φ.
Taking ψ = φ2 , we obtain a non-contractible cork (P, ψ) by [14]. Lemma
3.2 in [14] says that ψ induces the trivial map on the homology group.
Since (P, φ) changes any crossing for Fintushel-Stern’s knot-surgery, there
exists an embedding P ,→ E(2) such that the twist obtains E(2)(P, ψ n ) =E(2)T2n as proven in [14].
Suppose there exists a core Z-cork (D, f ) ⊂ (P, ψ) which (D, f k ) induces(P, ψ k ). Since D is a contractible, clearly P − D is not diﬀeomorphic toa cylinder. This setting says that this infinite order cork twist gives thefollowing twist:
E(2) ; E(2)(D, f k ) = E(2)T2k .

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MOTOO TANGE

However, the manifolds E(2)T2k have infinite OS-invariants with F-coeﬃcient.
This is a contradiction on Main theorem 1.
□
3. A core plug of (P, φ).
3.1. Proof of Proposition 1.10. The first picture (denoted by Q) in Figure 4 is P deleting an embedded disk. The diﬀeomorphism Figure 3 works

Figure 4. From the 2-handle deleted P to 0-framed 52
even for this submanifold Q. We denote the diﬀeomorphism ∂Q → ∂Q by
ϕ. Thus we obtain (Q, ϕk ) ⊂ (P, φk ) for any k.
Since the twisted double Q ∪ϕk (−Q) by (Q, ϕk ) is homeomorphic to
CP 2 #CP 2 (k: odd) and S 2 × S 2 (k: even) by easy calculation. The diffeomorphism ϕ2k+1 cannot extend to a homeomorphism on Q by [3]. Thus
(Q, ϕ2k+1 ) is a plug and (Q, ϕ2k ) is a non-contractible cork. Hence (Q, {ϕk })is a core Z-plug of (P, {φk }).
Therefore, for an unknotting number 1 knot K, there exists an embedding
Q ,→ E(2) such that E(2)(Q, ϕ) = E(2)K . Thus (Q, ϕ) is infinite order.
The handle diagram of Q can be reduced to 52 with framing 0. The
maximal Thurston-Bennequin invariant of 52 is 1. Thus the manifold is
Stein manifold. For example see Figure 1.
□
The presentation in Figure 1 is the famous Chekanov-Eliashberg knot.
Q ∪ϕ (−Q) and Q ∪ϕ2 (−Q) is diﬀeomorphic to CP 2 #CP 2 and S 2 × S 2
respectively.
Conjecture 3.1. For k ̸= 1, 2 any double Q ∪ϕk (−Q) is a standard 4manifold.
3.2. An action ϕ∗ on HF − (∂Q, sk ). Finally we compute the Heegaard
Floer homology of ∂Q and consider the action on the homology induced by
ϕ. Since 52 is an alternating knot, we have the following computation:
 2
i=1

F(0)
3
\
HF
K(52 , i) = F(−1) i = 0 2
F(−2) i = −1.

NON-EXISTENCE THEOREMS ON INFINITE ORDER CORKS

Now, the Heegaard Floer homology of −Σ(2, 3, 11) is as follows:+
HF + (S13 (52 )) = T(−2)⊕ F(−2) .

By using the surgery exact sequence in [10] among S 3 , S03 (52 ) = ∂Q, andS13 (52 ) = −Σ(2, 3, 11) we compute
{ +
+
k=0
T(− 1 ) ⊕ T(−
3 ⊕ F(− 3 )
+
)
2
2
2
HF (∂Q, sk ) =0k ̸= 0.
By using the exact sequence among HF − , HF ∞ , and HF + , we have thefollowing computation:{ −−T(− 5 ) ⊕ T(−
k=0
7 ⊕ F(− 5 )
−
)
2
2
2
HF (∂Q, sk ) ∼=F[U ]/(U k − 1)k ̸= 0.
The twist ϕ induces an action on HF − (∂Q) with spinc structures preserving,
because Spinc (∂Q) is equivalent to Spinc (∂Q × I) naturally. Here we state
the following question:
Question 3.2. How does the diﬀeomorphism ϕ on Heegaard Floer invariants
aﬀect?
To analyze the action would be significant to study the exotic structuresthat Fintushel-Stern’s knot surgery gives.
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Geom. & Top. 9(2015) 1-17

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MOTOO TANGE

Institute of Mathematics, University of Tsukuba, 1-1-1 Tennodai, Tsukuba,
Ibaraki 305-8571, Japan
E-mail address: tange@math.tsukuba.ac.jp