NOTES ON GOMPF’S INFINITE ORDER CORKS
MOTOO TANGE

Abstract. For any positive integer n we give a Zn -cork with a Zn eﬀective embedding in a 4-manifold being homeomorphic to E(n). Thismeans that a cork gives a subset Zn in the diﬀerential structures onE(n). Further, we describe handle decompositions of the twisted doubles(homotopy S 4 ) of Gompf’s infinite order corks and show that they are
Gluck twists and log transforms of S 4 .

1. Introduction
1.1. Twist and cork. Let X be a smooth manifold and Y a codimension 0
submanifold with a smooth embedding i : Y ,→ X. Removing Y from X andregluing by a self-diﬀeomorphism f : ∂Y → ∂Y , we obtain a new smoothmanifold and denote the manifold by X(i, Y, f ) or simply X(Y, f ). We callthe map f a gluing map or a twist map. This operation is called a twist anddenote it by (Y, f ). If the gluing map f extends to Y as a diﬀeomorphism,
then we call (Y, f ) a trivial twist.
Let Y be a contractible 4-manifold. We call a nontrivial twist (Y, f ) a
cork. Then the gluing map f is called a cork map. Corks play significant
roles in studying exotic 4-manifolds. ‘Exotic’ means that the manifolds are
homeomorphic but non-diﬀeomorphic each other. In fact, the following fact
is well-known.
Fact 1.1 ([7], [2]). Let X, X ′ be two closed simply connected exotic 4manifolds. Then there exists a cork (C, τ ) such that X ′ = X(C, τ ) andτ 2 = e.
1.2. Gompf ’s infinite order corks. Gompf in [4] gave an infinite exotic
family using infinite order corks as below. Let X be a certain 4-manifold(including a square zero torus with two vanishing cycles).
Fact 1.2 ([4]). Suppose that Kn is the 2n-twist knot. Then there exists an
infinite order cork (C, f ) satisfying XKn = X(C, f n ).
XK is the (Fintushel-Stern) knot-surgery of X by K. The cork order of acork is defined to be the minimal positive number whose power of the twist
map is a trivial twist. Higher order corks are known to exist by [10] and [1].
Date: January 25, 2019.
1991 Mathematics Subject Classification. 57R55, 57R65.
Key words and phrases. Infinite order cork.
The author is supported by JSPS KAKENHI Grant Number 26800031.
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Definition 1.3 (G-eﬀective embedding (defined in [1])). Let C be a 4manifold with boundary. Let G be a group acting on ∂C eﬀectively. If
there exists an embedding i of C into a 4-manifold X such that X(i, C, g)
is not diﬀeomorphic to X(i, C, g ′ ) for any distinct g, g ′ ∈ G, then we call theembedding i a G-eﬀective.
1.3. Galaxy. Here we give terminologies related to cork and cork twist.
These terminologies make it easier to understand our results. Let X be asmooth 4-manifold. We call the set of exotic structures on X the galaxy of
X and denote it bygal(X).
Our interest is to understand some kind of structures on the set gal(X). Any
cork twist can be regarded as some relationship among subsets of gal(X).
Let C be a contractible 4-manifold. Let G be a nontrivial subgroupin Diﬀ(∂C). Let (C, G) be a G-cork and C ,→ X a G-eﬀective embedding. Then the collection S = {X(C, g)|g ∈ G} is a subset of gal(X)with one to one correspondence g 7→ X(C, g). We call such a subset S∼=
a (G-)constellation and the embedding G → S ⊂ gal(X) a constellation
embedding.
Fact 1.1 means that a pair of every two points in gal(X) is a Z2 -constellation.
If G ,→ gal(X) is a constellation with respect to a G-cork (C, G), then any
subgroup e ̸= H < G gives an H-constellation H ,→ gal(X) with respect
to an H-cork (C, H). We call this constellation a subconstellation. The
main theorem in [11] says that any infinite family in gal(X) is not always aconstellation.
1.4. Results. In this subsecion, we digest the results (in Section 1.5 and1.6) obtained in this article.
The first result (Theorem 1) gives a construction of Zn -cork. Furthermore, we show that this cork gives a Zn -constellation in gal(E(n)) by knotsurgeries on a single fibered knot. This construction is due to an n-foldboundary sum of Gompf’s C.
The twisted double Sr,s,m,k of Gompf’s infinite order corks (C, f ) is given
by Gompf in [4]. We investigate the diﬀeomorphism type of Sr,s,m,k . We
prove Sr,s,m,k ∼= Sr,s,m+2,k . We prove Sr,s,0,k is diﬀeomorphic to the standard
4
S and Sr,s,1,k is the Gluck twists of S 4 along a 2-knot in S 4 . These are thesecond result (Theorem 2). The third result (Theorem 3) is a log transformconstruction of Sr,s,m,k . As a result, Sr,s,m,k is a (1/s)-log transform (or(−1/r)-log transform) along a torus in S 4 . As a result, Sr,s,m,k is a homotopy
S 4 having three kinds of constructions: a cork twist, a Gluck twist and alog transform.
1.5. Zn -corks. In [4] Gompf defined infinite order corks (C, f ) and asked in[4] whether there exists a Z2 -cork by taking the full T 2 action of his corks.
In [5] he partially gave a negative answer for this question. We construct a
Zn -cork below, but it is not an answer of this question.

NOTES ON GOMPF’S INFINITE ORDER CORKS

Theorem 1. For any natural number n there exists a Zn -cork Cn . Furthermore, there exist a 4-manifold Xn (homeomorphic to E(n)) and Zn -eﬀective
embedding Cn ,→ Xn . This Zn -eﬀective embedding gives a Zn -constellation
Zn ,→ gal(E(n)).
Here Cn is the boundary sum of n copies of C(1, 1; −1) as defined in [4].
This construction is due to performing cork twisting at distinct two clasps as
mentioned by Gompf in [4]. Here we give the following interesting questions:
Question 1.4. Is there a 4-manifold X such that gal(X) includes a Zn constellation for every natural number n?
Question 1.5. Does there exist a 4-manifold X with a G-constellation ingal(X) for an infinite non-abelian group?
Related topics to this question will be written in a sequel.
1.6. The twisted double of Gompf ’s (C, f ). Let C denote C(r, s; m).
The double D(C) = C ∪id (−C) is the boundary of C × I = B. Since, C isa Mazur type (consisting of one 1-handle and one 2-handle) contractible 4manifold, B is diﬀeomorphic to the 5-ball. Because, the attaching sphere of
the 5-dimensional 2-handle is S 1 in 4-space, it depends only on the homotopy
type of S 1 . Thus, the two handles are a canceling pair. In particular, thedouble D(C) is diﬀeomorphic to S 4 .
The twisted double C ∪f k (−C) = S 4 (C, f k ) =: Sr,s,m,k are all homotopy
4-spheres. As proven above, since the untwisted double Sr,s,m,0 is S 4 . The
problem that we concern is the diﬀeomorphism type of a general Sr,s,m,k .
Conjecture 1.6 ([4]). Let r, s, m be any integers with r, s > 0 > m. Let k
be a nonzero integer. Then Sr,s,m,k is standard S 4 .
We will prove the following theorem on Sr,s,m,k in Section 3.
Theorem 2. For any integer m, Sr,s,m,k ∼= Sr,s,m+2,k holds. If m = 0, thenSr,s,0,k is diﬀeomorphic to S 4 . If m = 1, then Sr,s,1,k is the Gluck twist along
a 2-knot Kr,s,k in S 4
We give another aspect of Sr,s,m,k .
Theorem 3. Sr,s,m,k is a (1/s)-log transform of S 4 along an embedded torus.
Note that exchanging the roles of r and s, we also know that Sr,s,m,k is(−1/r)-log transform of S 4 . From the proof of Theorem 3 we get a handle
decomposition of Sr,s,m,k .
Proposition 1.7. Sr,s,1,k admits a handle decomposition with one 0-handle,
two 1-handles, four 2-handles, two 3-handles, and one 4-handle.
If the embedding of the torus in S 4 extends to a fishtail neighborhood,then the log transform does not change the diﬀeomorphism type as discussed in [5]. Other similar situations appear in [12] and [13]. In [13], it isshown that the knot-surgery of S 4 along a torus is trivial by using a fishtail
neighborhood embedding in S 4 . It, however, seems diﬃcult to find a certain

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fishtail neighborhood in Sr,s,m,k . Distinguishing the diﬀerential structuresof Sr,s,m,k might be a hard work.
Sr,s,m,k is also considered as a 4-manifold obtained by log transforms along
two tori in S 4 as written in Section 3.1. The tori have a symmetry, then we
have Sr,s,m,k ∼= S−s,−r,m,k due to the handle diagram in Figure 10.
Furthermore, the 0-log transform and (0, 0)-log transform of S 4 withrespect to the torus and the tori are homotopy S 3 × S 1 #S 2 × S 2 and#2 S 3 × S 1 #2 S 2 × S 2 respectively. By the similar argument to the proof of
Theorem 2, if m is even, then S∞,s,m,k ∼= S 3 × S 1 #S 2 × S 2 and S∞,∞,m,k ∼=#2 S 3 ×S 1 #2 S 2 ×S 2 hold. In the case where m is odd, S∞,s,m,k and S∞,∞,m,k
are the Gluck twists along the standard 4-manifolds.
Question 1.8. Are S 4 [0] and S 4 [0, 0] diﬀeomorphic to the standard (S 3 ×
S 1 )#(S 2 × S 2 ) or #2 (S 3 × S 1 )#2 (S 2 × S 2 ) respectively?
A similar construction is a Scharlemann manifold in [8], which is a surgeryof Σ × S 1 for a rational homology sphere Σ. The surgeries are done alongnormally generating loops of π1 (Σ) in Σ × S 1 . In the case where Σ is a Dehn
surgery of a knot and the loop is the meridian of the knot, the manifold isequivalent to a knot-surgery of the double of the fishtail neighborhood. For
example see [13]. The general Scharlemann manifold gives a homotopy#(S 3 × S 1 )#l (S 2 × S 2 )#m (CP 2 #CP 2 ).
3 (left handed 3 ) and the loop is the
The case where Σ = Σ(2, 3, 5) = S−1
1
meridian of the trefoil is the original one in [8]. The author in [13] provedthat some Scharlemann manifolds are standard. Here we give another question.

Question 1.9. Are S 4 [0] and S 4 [0, 0] diﬀeomorphic to a Scharlemann manifold and a connected-sum of those manifolds respectively?
Acknowledgements
This study here was inspired by [4] and partially done during my stayat KIAS on March in 2016. I am deeply grateful for the hospitality at
the institute, and Kyungbae Park and Min Hoon Kim. Kouichi Yasui gaveme much advice and suggestions in the Hiroshima University TopologyGeometry seminar. Also, Robert Gompf gave me much useful comments
and helps for this earlier manuscript of this paper. I thank them so much.
2. Gompf’s infinite order cork C.
2.1. Knot-surgery. Before drawing Gompf’s infinite order corks, we review the definition of Fintushel-Stern knot-surgery. Let K be a knot in S 3 .
Let X be a 4-manifold with a square zero embedded torus T . Then theoperation
XK = [X − ν(T )] ∪ [(S 3 − ν(K)) × S 1 ]NOTES ON GOMPF’S INFINITE ORDER CORKS

is called a (Fintushel-Stern) knot-surgery along K. The gluing map is definedin [3]. The notation ν(·) stands for a tubular neighborhood of a submanifold.
Then the Seiberg-Witten invariant formula of knot-surgery is the following:
SWXK = SWX · ∆K (t),where ∆K (t) is the Alexander polynomial of K.
We remark Sunukjian’s work in [9]. The paper says that if a 4-manifold
X has a non-trivial Seiberg-Witten invariant, then two knot-surgeries XK1and XK2 along knots K1 and K2 with ∆K1 ̸= ∆K2 cannot be diﬀeomorphic. In short, Alexander polynomial distinguishes smooth structures ofknot surgeries.
2.2. A handle diagram of C. In this section we describe the handle diagram of Gompf’s infinite order cork C. We call a handle diagram a diagram
simply. In [6] the diagram is described by the diﬀerent method. The manifold C = C(r, s; m) can be built by attaching a single 2-handle to I × P ,where P is the complement of (r, s)-double twist knot κ(r, −s) as in Figure 1 in [4].
Lemma 2.1. The diagram of C is Figure 1 and the cork map f is Figure 2.

Figure 1. The handle decomposition of C(r, s; m).
The boundary of Gompf’s contractible 4-manifold C has a torus decomposition along an incompressible torus. The incompressible torus can berealized as a torus indicated in Figure 1. Note that in the diagram in
Figure 1 the torus apparently cannot be realized as an embedded torus,
because the torus meets the dotted 1-handles 4 times. However, by inserting
2 pairs of canceling 2/3-handles, and putting the torus over the 2-handle, we
can avoid the intersections. As a result we can find our required embedded
torus.
Proof.
Let Σ be a punctured torus. Σ × S 1 is diﬀeomorphic to the
0-surgered solid torus along the Bing double as in the left of Figure 3.
Thus by the fundamental method of handle calculus, the picture of thecylinder I ×Σ×S 1 is the right diagram in Figure 3. Attaching a −r-framed

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Figure 2. The diﬀeomorphism f on ∂C(r, s; m).

Figure 3. Σ × S 1 and Σ × S 1 × I.

2-handle (an s-framed 2-handle on another side respectively) and removing
the union of the core and the attaching sphere cross interval I, we obtain
the other handle attachment and removal in Figure 1. The two 0-framed

NOTES ON GOMPF’S INFINITE ORDER CORKS

2-handles in the right of Figure 3 are canceled out with the 3-handles whenremoving the cores. The similar situation appears in [12].
Hence, the handle diagram of C is as in the picture in Figure 1. Reducing
the diagram, we get a ribbon 1-handle and m-framed 2-handle along themeridian of the ribbon knot.
□
The map f is defined to be the right handed Dehn twist cross the identityon(I × ∂Σ) × S 1 .
This cork C produces a knot-surgery XK for a 4-manifold X for a squarezero torus T . The torus satisfies the following condition.
We assume r = s = −m = 1. Let V be the neighborhood of the Kodaira’s
singular fibration of type III, which V is also used in [14] in the same situation. Suppose that V is embedded in X. T is embedded as a general fiberof V ⊂ X. Then there exists an embedding C ⊂ V ⊂ X such that
XK = X(C, f k ),where K = κ(k, −1) is the 2k-twist knot.
If C does not satisfy r = s = −m = 1, we can construct the knot surgery
by the twist knot on an embedded torus under certain condition as mentioned in [4].
Due to [4], for any integer k the k-fold composition f k cannot extend to
the inside C as any diﬀeomorphism.
We describe in Figure 4 the local deformation of X(C, f 2 ) according to
the diﬀeomorphism f in Figure 2.

Figure 4. The cut and paste of C(r, s; m) by f 2 .

2.3. 2-bridge knots Km,n . In the next section we prove Theorem 1. First
we prepare a 2-bridge knot Km,n as in Figure 5 for integers m, n. The knot

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Figure 5. Km,n .
Km,n is classified as follows:T2,2m−1unknot(1)
Km,n =

T2,−3non-torus 2-bridge knot

n=0(m, n) = (−1, −1)(m, n) = (0, 1)
otherwise.

The following equality holds
K−1,n ≈ K0,n+1 .
Here Tp,q is the right handed (|p|, |q|)-torus knot if pq > 0 and is the left
handed (|p|, |q|)-torus knot if pq < 0.
Let ∆m,n denote the Alexander polynomial ∆Km,n . The Alexander polynomial is computed as follows: If m ≥ 1, then.
∆m,n (t) = n(tm+1 + t−m−1 ) − 3n(tm + t−m ) + (4n + 1)

m−1∑(−1)i−m+1 ti

i=−m+1and if m = 0, then∆0,n (t) = n(t + t−1 ) − (2n − 1).
This formula can be easily proven by using the skein relation of the Alexander polynomial.
We prove the following lemma:
Lemma 2.2. If m ≥ 1, then the polynomials ∆m,n (t) are distinct each otherin Z[t, t−1 ]/ ± t±1 . In particular, for (m, n) ̸= (m′ , n′ ) with m, m′ ≥ 1, Km,nand Km′ ,n′ are non-isotopic as unoriented knots.
Proof.
Suppose that ∆m,n = ∆m′ ,n′ . If n ̸= 0 and n′ ̸= 0 hold,then, comparing the coeﬃcients of top degree terms, we have n = n′ andm = m′ . If either n or n′ is 0 and the other is not 0, then the two polynomials do not agree. Suppose that n′ ̸= 0 and n = 0 hold. Then we have∆m,n (t) = ∆T2,2m−1 (t). Comparing the top degrees of ∆m,n (t) and ∆m′ ,n′ (t),we have n′ = ±1. However, comparing the values of the second top degreeof ∆m′ ,n′ (t) we must have n′ = ±3. This is contradiction. If n = n′ = 0,
then, comparing the top degrees of the two polynomials, we have m = m′ .
2

NOTES ON GOMPF’S INFINITE ORDER CORKS

2.4. Proof of Theorem 1. We remark that the natural number k in thisproof corresponds to n in the statement.
Let K(n1 , · · · , nk ) be K1,n1 # · · · #Kk,nk . Hence K(0, · · · , 0) = #ki=1 T2,2i−1
holds. We embed k copies of Σ in the exterior E of K(0, · · · , 0) disjointly.
In the case of 2 copies see Figure 6. Thus, disjoint k copies of I × Σ × S 1

Figure 6. A disjoint embedding of two copies of Σ in E2 .
in E × S 1 are also embedded. By attaching 2k (−1-framed) 2-handles on
the meridians of E × S 1 and k (−1-framed) 2-handles on the S 1 -direction,
we can embed k copies of C = C(1, 1; −1) in Xk := E(k)#k T2,2i−1 , because
i=1
it has 12k vanishing cycles.
We take each point in the complements of the incompressible tori in∂C ∪ ∂C and connect the two components by a 1-handle attached on theneighborhoods of the two points in Xk , which the 1-handle is disjoint fromk C’s. Embedding such k − 1 1-handles, we construct ♮k C ,→ Xk .
Let fi be a diﬀeomorphism on ∂(♮k C) which acts as Gompf’s f on thei-th component of #k ∂C and acts as the identity on the other component of#k ∂C. Since those points are taken in the complement of the incompressible
tori, the two maps fi and fj are commutative. The twist (♮k C, f1n1 · · · fknk )of Xk produces E(k)K(n1 ,··· ,nk ) .
Then, the computation of the Seiberg-Witten invariants is as follows:
SWE(k)K(n ,··· ,n ) = SWE(k)
1

k
∏

k

∆i,ni

i=1

SWXk = SWE(k)k∏∆i,0 .

i=1

Comparing the degrees of the two results, the two Seiberg-Witten invariants
do not agree, unless ni = 0 for all i. Since Xk and E(k)K(n1 ,··· ,nk ) are exotic when (n1 , · · · , nk ) ̸= (0, · · · , 0). (♮k C, f1n1 · · · fknk ) gives an exotic E(k).
Thus, (♮k C, f1n1 · · · fknk ) is a cork. This means that (♮k C, {f1n1 · · · fknk |nj ∈Z}) is a Zk -cork.
To prove that this embedding is a Zk -eﬀective embedding, we have only to∏∏show that if ki=1 ∆i,ni (t) = ki=1 ∆i,n′i (t), then (n1 , · · · , nk ) = (n′1 , · · · , n′k ).

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Claim 2.3. Let k, p be integers with k > 0 and p ≥ 0 and ni , n′i (i = 1, · · · , k)integers. If we have
k−p
∏

(2)

∆p+i,np+i =

i=1

k−p
∏

∆p+i,n′p+i ,i=1

then we have np+1 = n′p+1 .
If ∆p,q were irreducible, then this claim would be easy. However, sincesome 2-bridge knots are ribbon, such Alexander polynomials are not always
irreducible.
Proof. By the induction of the number k in (2) we prove this claim.
Let σi and σi′ be the i-th elementary symmetric polynomials in np+1 , · · · , nk
and n′p+1 , · · · , n′k respectively. For example, we see∑
σi =nℓ1 · · · nℓi .
{ℓ1 ,··· ,ℓi }⊂{p+1,··· ,k},#{ℓ1 ,··· ,ℓi }=i

Let d be the degree of (2). Comparing the degree d of (2), we obtain′.
σk−p = σk−p

Let j0 be an integer with 1 ≤ j0 ≤ p + 1. Comparing the coeﬃcients of the′degree d − 2j of (2) with 0 ≤ j ≤ j0 we have σk−p−j0 = σk−p−j.
0Further, the coeﬃcients with degree d − 2p − 3 of the left hand side of (2)
is
k−p
∏

S + (−3np+1 )

j=p+2

nj +

k−p
∑

(−4nj − 1)

j=p+2

k
∏

nℓ

ℓ=p+1

ℓ̸=j

= S0 −k∑k∏nℓ = S0 − σk−p−1 + np+2 · · · nk ,j=p+2 ℓ=p+1ℓ̸=jwhere S, S0 are polynomials generated by σk−p , σk−p−1 · · · , σk−p−j .
′, we haveThus np+2 · · · nk = n′p+2 · · · n′k holds. By using σk−p = σk−p′np+1 = np+1 .
□∏k
We go back to the proof of Theorem 1. Suppose that i=1 ∆i,ni =∏k′i=1 ∆i,n′i . Then, by using Claim 2.3 in the case of p = 0 we have n1 = n1 .
∏k
By dividing ∆1,n1 from the both side of this equality, we have i=2 ∆i,ni =∏k′i=2 ∆i,n′i . Iterating this process by using Claim 2.3, we have n2 = n2 , · · · ,nk = n′k holds.
∏∏
Thus ki=1 ∆i,ni = ki=1 ∆i,n′i implies (n1 , · · · , nk ) = (n′1 , · · · , n′k ). Therefore we show that this embedding♮k C := Ck ,→ Xk = E(k)#k

i=1 T2,2i−1

is Zk -eﬀective.

□

NOTES ON GOMPF’S INFINITE ORDER CORKS

This proof focuses on the case of C = C(1, 1; −1). However, Gompf’s
method in Remarks (a) in [4] would change our examples to C = C(r, s; m)with r, s > 0 > m.
3. The twisted double Sr,s,m,k of C.
3.1. The diagrams of twisted doubles. Let Sr,s,m,k denote the homotopy
S 4 defined in Section 1.6. We prove the following proposition first of all.
Proposition 3.1. In the case of k = 1, the diagram of Sr,s,m,1 is Figure 9.
The handle diagram of Sr,s,m,k is Figure 10.
Proof.
The move of the meridians of 2-handles by f is described in
Figure 7. Hence by f , the meridians of four 2-handles in Figure 1 are

Figure 7. The move of meridians of 2-handles by the f .
moved to the link α, β, γ and δ in Figure 9.
Hence, this diagram in Figure 9 is the k = 1 case. In the general kcase, the images of the meridians of the 2-handles by f k are α, β, γ and δ in
Figure 10. Thus the first diagram in Figure 10 describes Sr,s,m,k . The β, γand the meridian of the m-framed knot can be canceled with two 3-handles,
because the calculation in Figure 8 proves that the three components are
the unlink. After the canceling, β, γ and the meridian of the m-framed
2-handle are the 0-framed unlink on the handle decomposition of the 2handlebody. Thus we cancel those together with three 3-handles. Then we
obtain the second picture as in Figure 10.
□
Proof of Proposition 1.7. The union of 2-handles α, δ, two 3-handles
and a 4-handle is diﬀeomorphic to C. Hence Sr,s,m,k = C ∪f k (−C) admits
two 1-handles and four 2-handles.
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Figure 8. This calculation proves that the red three components is the unlink.

Figure 9. Twisted double Sr,s,m,1 .

In particular, in Figure 10, the four 2-handles are α, δ, s-framed 2-handle
and (−r)-framed 2-handle.

NOTES ON GOMPF’S INFINITE ORDER CORKS

Figure 10. Twisted double Sr,s,m,k and a diagram after canceling.

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If one proves that Sr,s,m,k is standard, the first diagram in Figure 10
would be useful as an auxiliary information of the handle decomposition.
Let X be a 4-manifold. We consider an embedded S 2 ,→ X with trivial
normal bundle. The homotopy types of self-diﬀeomorphisms on S 2 × S 1 isisomorphic to Z/2Z. Let τ be a self-diﬀeomorphism over S 2 × S 1 with nontrivial homotopy class. The surgery X ; X(S 2 × D2 , τ ) is called a Gluck
twist.
3.2. Proofs of Theorem 2. First we prove that Sr,s,0,k is diﬀeomorphic to
S 4 . The deformation of the pictures in Figure 17 presents the Sr,s,0,k ∼= S4.
In the diagram of Sr,s,0,k the union of the (lowest) 0-framed 2-handle anda 4-handle consists of S 2 × D2 . By removing the S 2 × D2 and attaching an
m-framed 2-handle and a 4-handle, we obtain the surgery S 4 = Sr,s,0,k ;Sr,s,0,k (S 2 × D2 , τ m ) = Sr,s,m,k . Thus Sr,s,m,k is the Gluck surgery of S 4 .
Hence, the diﬀeomorphism Sr,s,m,k ∼□= Sr,s,m−2,k holds.
The last diﬀeomorphism can be also verified by the calculus in Figure 11.

Figure 11. The diﬀeomorphism Sr,s,m,k ∼= Sr,s,m−2,k .

3.3. The log transform along several tori. Here we define the log transform along n tori with the square zero. Let ei : Ti2 ,→ X be disjoint embedded tori each other that the squares are all zero for i = 1, · · · n. Let ci
be a curve presenting a primitive element in H1 (Ti2 ). Let pi , qi be several
pairs of coprime integers. Let ẽi be an embedding of the tubular neighborhood of T 2 with respect to ei . Suppose that a gluing diﬀeomorphism
gci ,pi ,qi : T 2 × ∂D2 → Ti2 × ∂D2 a diﬀeomorphism satisfying∂D2 7→ pi · ∂D2 + qi · ci .
In fact, the image of ∂D2 by the gluing map is the attaching sphere of theunique 2-handle in T 2 × D2 and the remaining handles of T 2 × D2 are two 3handles and one 4-handle. The information of the image of ∂D2 determinesa diﬀeomorphism type. Hence, the diﬀeomorphism type of the log transform
along the tori depends only on ei , ci , pi and qi .
We denote the log transform by
X[(ei ), (ci ), (pi /qi )].
and call (pi /qi )-log transform along (ei ) with direction (ci ). When embeddings (ei ) and curves (ci ) are clear in the context, we omit these items.

NOTES ON GOMPF’S INFINITE ORDER CORKS

3.4. Proof of Theorem 3. First, we find T 2 × D2 in the diagrams in
Figure 9 (Sr,s,m,1 ), in general, Figure 10 (Sr,s,m,k ). Removing handles, we
can show that the submanifold as in Figure 12 is embedded in Sr,s,m,k . It
contains two disjoint T 2 s in the last picture.

Figure 12. Disjoint embedded two T 2 ’s in S 4 .
One time (1/1)-log transform with direction ℓ corresponds to the change ofthe diagram which is given in Figure 13. In general, the (1/s)-log transformis the s-times iteration of this process. Hence, Sr,s,m,k is the (1/s)-log trans-

Figure 13. The (1/1)-log transform with direction ℓ.
form of Sr,0,m,k . Since the knot κ(r, 0) is isotopic to the unknot, C(r, 0; m)
is the standard 4-ball. Thus Sr,0,m,k is diﬀeomorphic to S 4 .
Therefore, Sr,s,m,k is a (1/s)-log transform along a torus embedded in S 4 .
By exchanging the roles of r and s, we can show that Sr,s,m,k is a (−1/r)-log
transform along another torus in S 4 .
□Theorem 3 says that the twisted double Sr,s,m,k is obtained by two logtransforms along two embedded disjoint tori in S 4 as in Figure 12. Namely,
we haveSr,s,m,k = S 4 [(em,k,1 , em,k,2 ), (c1 , c2 ), (−1/r, 1/s)] = S 4 [(−1/r, 1/s)].

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The two torus embeddings em,k,i : Ti2 ,→ S 4 (i = 1, 2) are embedded in sucha way that each torus is embedded in each component T 2 × D2 in ♮2 T 2 × D2 .
The T 2 × D2 ♮T 2 × D2 exterior in S 4 is described in Figure 14.

Figure 14. The ♮2 T 2 × D2 exterior in S 4 and curves c1 , c2
on the boundary.

3.5. A remark for the curves c1 and c2 . As mentioned in Section 1.6,
if either of curves c1 or c2 in the boundary has an embedded slice disk inthe exterior with framing −1. However, it is diﬃcult to find such a disk interms of the following observation.
We consider a cobordism C from #2 T 2 × S 1 to #3 S 2 × S 1 by removing
the three 3-handles and one 4-handle from the exterior which is describedin Figure 14. We take an annulus in C beginning from either of c1 or c2 .
Suppose that the annulus has no critical points in C, i.e., the annulus is thetrace by the gradient flow of the Morse function for the handle decomposition. Let c̃ be the obtained knot in #3 S 2 × S 1 . Turning the union of three
3-handles and a 4-handle by the upside down calculus, we obtain a knot
description in Figure 15. Clearly, this knot c̃ has no −1-framed disk in♮3 D3 × S 1 . Because if there is such a disk, then by attaching a 2-handle on c̃with 0-framing, we must find a (−1)-sphere in the attached manifold whose
intersection form is ⟨0⟩. This is a contradiction.
This means that the easy way cannot be found a −1-framed embedded
disk in the exterior for em,k,i .
3.6. The cases of r or s = ∞. The cases where either r or s is ∞ canbe regarded as 0-log transforms along the tori from the equality Sr,s,m,k =S 4 [−1/r, 1/s]. Namely, S∞,s,m,k = S 4 [0, 1/s] and S∞,∞,m,k = S 4 [0, 0].
S∞,∞,m,k is obtained by exchanging two dots and two 0’s in the subhandle for ♮2 T 2 × D2 . The diagram of S∞,∞,m,k is described in Figure 16.

NOTES ON GOMPF’S INFINITE ORDER CORKS

Figure 15. The isotopy class of c1 ⊂ ∂(♮3 D3 × S 1 ).
By computing the fundamental groups and homology groups, the manifolds
S∞,0,m,k and S∞,∞,m,k are homotopic to(S 3 × S 1 )#(S 2 × S 2 ) and #2 (S 3 × S 1 )#2 (S 2 × S 2 )respectively. In the case of m ≡ 0 mod 2, by using the same move as
Figure 17, we have S∞,s,m,k = (S 3 × S 1 )#(S 2 × S 2 ) and S∞,∞,m,k =#2 (S 3 × S 1 )#2 (S 2 × S 2 ). Thus essentially, we have to investigate the
manifold S∞,s,1,k and S∞,∞,1,k . These are the Gluck twists of standard(S 3 × S 1 )#(S 2 × S 2 ) or #2 (S 3 × S 1 )#2 (S 2 × S 2 ).
In [13] some homotopy (S 3 × S 1 )#(S 2 × S 2 )’s are constructed according
to Scharlemann’s method. Some of those are diﬀeomorphic to the standard
manifold. What one knows the relationship between these manifolds would
be an interesting problem.

Figure 16. The handle diagram of S 4 [0, 0].

References[1] D. Auckly, H. Kim, P. Melvin, and D. Ruberman, Equivariant corks, Algeb. Geom.
Topol. 17 (2017) 1771–1783.

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Figure 17. Sr,s,0,k is the standard S 4 .

[2] 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),
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[3] R. Fintushel and R. Stern, Knots, links and 4-manifolds, Invent. Math. 134 (1998),
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[4] R. Gompf, Infinite order corks, Geom. Topol. Volume 21, Number 4 (2017), 2475–
2484.
[5] R. Gompf, More Cappell-Shaneson spheres are standard, Algebr. Geom. Topol.10(2010), no. 3, 1665–1681.
[6] R. Gompf, Infinite order corks via handle diagrams, Algebr. Geom. Topol. Volume
17, Number 5 (2017), 2863–2891.
[7] R. Matveyev, A decomposition of smooth simply-connected h-cobordant 4-manifolds,
J. Diﬀ. Geom. Vol. 44 (1996) 571–582.
[8] M. Scharlemann, Constructing strange manifolds with dodecahedral space, Duke Math.
J 43(1976) 33–40.
[9] N. S. Sunukjian, A note on knot surgery. J. Knot Theory Ramifications 24 (2015),
no. 9, 1520003, 5 pp.
[10] M. Tange, Finite order corks, Internat. J. Math. 28 (2017), no. 6, 1750034, 26 pp.
[11] M. Tange, Non-existence theorems on infinite order corks, arXiv:1609.04344.

NOTES ON GOMPF’S INFINITE ORDER CORKS[12] M. Tange, On Nash’s 4-sphere and Property 2R, Turkish J. Math. 37 (2013), no. 2,
360-374.
[13] M. Tange, The link surgery of S 2 ×S 2 and Scharlemann’s manifolds, Hiroshima Math.
J. 44 (2014), no. 1, 35–62.
[14] M. Tange, A plug with infinite order and some exotic 4-manifolds, Journal of Gökova
Geometry Topology - Volume 9 (2015) 1–17.
Institute of Mathematics, University of Tsukuba, 1-1-1 Tennodai, Tsukuba,
Ibaraki 305-8571, Japan
Email address: tange@math.tsukuba.ac.jp19