MOTOO TANGE Abstract. For any positive integer n we give a Z n -cork with a Z n - effective embedding in a 4-manifold being homeomorphic to E ( n ). This means that a cork gives a subset Zn in the differential structures on E(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. I NTRODUCTION
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 and regluing by a self-diffeomorphism f : ∂Y → ∂Y , we obtain a new smooth manifold and denote the manifold by X(i,Y,f) or simply X(Y,f). We call the map f a gluing map or a twist map. This operation is called a twist and denote 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 4- Then there exists a cork (C,T) such that X' = X(C,T) and manifolds. T 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 K n is the 2n-twist knot. Then there exists an infinite order cork (C, f) satisfying X κ n = X(C, f n ). X K is the (Fintushel-Stern) knot-surgery of X by K . The cork order of a cork 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].
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Definition 1.3 (G-effective embedding (defined in [1])). Let C be a 4- manifold 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 the embedding 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 a smooth 4-manifold. We call the set of exotic structures on X the galaxy of X and denote it by
gal( 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 subgroup in Diff(∂C).  Let (C,G) be a G-cork and C → X a G-effective embed- ding. Then the collection S = {X(C,g)|g ∈ G} is a subset of gal(X) with one to one correspondence g → 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 Z 2 -constellation. If G → gal(X) is a constellation with respect to a G-cork (C,G), then any subgroup e̸= 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 a constellation. 1.4. Results. In this subsecion, we digest the results (in Section 1.5 and 1.6) obtained in this article. The first result (Theorem 1) gives a construction of Z n -cork. Further- more, we show that this cork gives a Tn-constellation in gal( E ( n )) by knot- surgeries on a single fibered knot.  This construction is due to an n -fold boundary sum of Gompf's C . The twisted double S r,s,m,k of Gompf's infinite order corks ( C, f ) is given by Gompf in [4]. We investigate the diffeomorphism type of S r,s,m,k • We prove S r,s,m,k = Sr,s,m+2,k . We prove S r,s,0,k is diffeomorphic to the standard S 4 and S r.s.1.k is the Gluck twists of S 4 along a 2-knot in S 4 . These are the second result (Theorem 2). The third result (Theorem 3) is a log transform construction of S r,s,m,k . As a result, S r,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 a log transform. 1.5. Z n -corks. In [4] Gompf defined infinite order corks ( C , f ) and asked in [4] whether there exists a Z 2 -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. 2
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Theorem 1. For any natural number n there exists a Z n -cork C n . Further- more, there exist a 4-manifold X n (homeomorphic to E(n)) and Z n -effective embedding C n → X n .  This Z n -effective embedding gives a Z n -constellation Zn → gal(E(n)). Here C n 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 Z n - constellation for every natural number n ? Question 1.5. Does there exist a 4-manifold X with a G-constellation in gal(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 U id (−C) is the boundary of C × I = B. Since, C is a Mazur type (consisting of one 1-handle and one 2-handle) contractible 4- manifold, 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, the double D( C ) is diffeomorphic to S 4 . The twisted double C ∪ fk (−C) = S 4 (C, fk) =: 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 S4. 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, then Sr,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 54 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 S r.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. S r.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 dis- cussed in [5]. Other similar situations appear in [12] and [13]. In [13], it is shown 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 3
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fishtail neighborhood in Sr.s.m.k . Distinguishing the differential structures of Sr,s,m,k might be a hard work. S r.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 S r,s,m,k = S –s,–r,m,k due to the handle diagram in F IGURE 10. Furthermore, the 0-log transform and (0,0)-log transform of S r4 with respect 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 × $¹)#($² × $²) or #²($³ × $¹)#²($² × $²) respectively? A similar construction is a Scharlemann manifold in [8], which is a surgery of Σ × S 1 for a rational homology sphere Σ . The surgeries are done along normally generating loops of π 1 (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 is equivalent to a knot-surgery of the double of the fishtail neighborhood. For example see [13]. The general Scharlemann manifold gives a homotopy #(S³ × S¹)# l (S² × S²)#™(CP²#CP²). The case where Σ = Σ(2,3,5) = S 31 (left handed 31 ) and the loop is the meridian of the trefoil is the original one in [8]. The author in [13] proved that some Scharlemann manifolds are standard. Here we give another ques- tion. Question 1.9. Are S 4 [0] and S 4 [0,0] diffeomorphic to a Scharlemann man- ifold and a connected-sum of those manifolds respectively?
A A CKNOWLEDGEMENTS
This study here was inspired by [4] and partially done during my stay at KIAS on March in 2016.  I am deeply grateful for the hospitality at the institute, and Kyungbae Park and Min Hoon Kim. Kouichi Yasui gave me much advice and suggestions in the Hiroshima University Topology- Geometry 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 re- view 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 the operation
4
NOTES ON GOMPF’S INFINITE ORDER CORKS
is called a (Fintushel-Stern) knot-surgery along K . The gluing map is defined in [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 · ΔK(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 X K 1 and X K 2 along knots K 1 and K 2 with Δ K 1̸= Δ K 2 cannot be diffeomor- phic.  In short, Alexander polynomial distinguishes smooth structures of knot surgeries. 2.2. A handle diagram of C . In this section we describe the handle dia- gram 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 man- ifold 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 F IG - URE 1 in [4]. Lemma 2.1. The diagram of C is Figure 1 and the cork map f is Fig- URE 2. The boundary of Gompf's contractible 4-manifold C has a torus decom- position along an incompressible torus. The incompressible torus can be realized 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 the cylinder I × Σ × S 1 is the right diagram in F IGURE 3. Attaching a − r -framed 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 when removing 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 the meridian of the ribbon knot.口The map f is defined to be the right handed Dehn twist cross the identity on
This cork C produces a knot-surgery X K for a 4-manifold X for a square zero 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 situ- ation. Suppose that V is embedded in X . T is embedded as a general fiber of V ⊂ X. Then there exists an embedding C ⊂ V ⊂ X such that
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 men- tioned in [4]. Due to [4], for any integer k the k -fold composition fk 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. 2.3. 2-bridge knots K m,n . In the next section we prove Theorem 1. First we prepare a 2-bridge knot K m,n as in F IGURE 5 for integers m, n. The knot 7
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Km,n is classified as follows:
K-1,n ~ K0,n+1. Here T p,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 ∆ κ m,n . The Alexander poly- nomial is computed as follows: If m ≥ 1, then
and if m = 0, then
This formula can be easily proven by using the skein relation of the Alexan- der polynomial. We prove the following lemma: Lemma 2.2. If m ≥ 1, then the polynomials ∆m,n(t) are distinct each other in Z[t,t-1]/±t±1. In particular, for (m,n) ≠ (m',n') with m,m' ≥ 1, Km,n and Km',n' are non-isotopic as unoriented knots. Proof. Suppose that ∆ m,n = ∆ m ′,n ′ . If n̸= 0 and n ′̸= 0 hold, then, comparing the coefficients of top degree terms, we have n = n ′ and m = m ′ . If either n or n ′ is 0 and the other is not 0, then the two poly- nomials do not agree. Suppose that n ′̸= 0 and n = 0 hold. Then we have ∆m,n(t) = ∆ T 2,2m–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 degree of △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 ′ . □
NOTES ON GOMPF’S INFINITE ORDER CORKS
2.4. Proof of Theorem 1. We remark that the natural number k in this proof corresponds to n in the statement. Let K (n 1 , · · · ,n k ) be K 1 ,n 1 # · · · # K k,n k · Hence K (0 , · · · ,0) = # i = 1 T 2,2i – 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 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 X k := E ( k )# k − 1 , T 2 , 2 i − 1 , because 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 the neighborhoods of the two points in X k , which the 1-handle is disjoint from k C's. Embedding such k − 1 1-handles, we construct bkC → X k . Let f i be a diffeomorphism on ∂( i k C ) which acts as Gompf's f on the i-th component of #k∂C and acts as the identity on the other component of #kdC . Since those points are taken in the complement of the incompressible tori, the two maps f i and f j are commutative. The twist ( b × C, f j x 1 · · · f k k of Xk produces E(k)K(n1,…,nk) Then, the computation of the Seiberg-Witten invariants is as follows:
Comparing the degrees of the two results, the two Seiberg-Witten invariants do not agree, unless n i = 0 for all i. Since X k and E ( k ) K ( n i,..,n k ) are ex- otic when ( n 1 , · · · , n k )̸= (0 , · · · , 0). ( b k C , f l 1 · · · f l k ) gives an exotic E ( k ). Thus, ( t k C, f n 1 · · · f k k ) is a cork. This means that ( t k C, { f n 1 · · · f k k | n j ∈ Z}) is a Zk-cork. To prove that this embedding is a Zk-effective embedding, we have only to show that if Π i=1 △ i,n i (t) = Π i=1 △ i,n i (t), then (n 1, · · · ,n k) = (n i 1, · · · ,n i k).
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Claim 2.3.  Let k,p be integers with k > 0 and p ≥ 0 and n i,n i (i = 1, . . . ,k) integers. If we have
then we have np+1 = n'p+1 If ∆ p.q were irreducible, then this claim would be easy. However, since some 2-bridge knots are ribbon, such Alexander polynomials are not always irreducible. By the induction of the number k in (2) we prove this claim. Proof. Let σ i and σ i be the i-th elementary symmetric polynomials in n p+1, · · · ,n k and n p +1, · · · , n k respectively. For example, we see
Let d be the degree of (2). Comparing the degree d of (2), we obtain
Let j 0 be an integer with 1 ≤ j 0 ≤ p + 1. Comparing the coefficients of the degree d − 2j of (2) with 0 ≤ j ≤ jo we have σκ−ρ−jo = σk−ρ−jo Further, the coefficients with degree d — 2p — 3 of the left hand side of (2) is
where S , S 0 are polynomials generated by σk –p,σk –p –1 · · · ,σk –p –j· Thus np+2 · · · nk = n p+2 · · n k holds. By using σ k – p = σ k – p , we have Np+1 = n'p+1 We go back to the proof of Theorem 1.  Suppose that Π i=1 Δ i,n i Π i=1 △ i,n ′ . Then, by using Claim 2.3 in the case of p = 0 we have n 1 = n ′ . By dividing △ 1 , n 1 from the both side of this equality, we have Π i = 2 △ i,n 1 = Π i=2 △ i,n i . Iterating this process by using Claim 2.3, we have n 2 = n l 2 , · · · , nk = n'k holds. Thus Π i =1 Δ i,n i = Π i =1 Δ i,n i implies (n 1, · · · ,n k ) = (n ′ 1, · · · ,n ′ k ). There- fore we show that this embedding
is Z k -effective. 10
□
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 S r,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. The move of the meridians of 2-handles by f is described in Proof. F IGURE 7. Hence by f , the meridians of four 2-handles in F IGURE 1 are 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 k case, the images of the meridians of the 2-handles by fk are α,β,γ and δ in F IGURE 10. Thus the first diagram in F IGURE 10 describes S r,s,m,k . The β, γ and 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 2- handlebody. Thus we cancel those together with three 3-handles. Then we obtain the second picture as in FIGURE 10.口Proof of Proposition 1.7. The union of 2-handles α, δ, two 3-handles and a 4-handle is diffeomorphic to C . Hence S r,s,m,k = C U fk (−C) admits two 1-handles and four 2-handles. → MOTOO TANGE In particular, in Figure 10, the four 2-handles are α, δ, s-framed 2-handle and (−r)-framed 2-handle. 12 13
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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 is isomorphic to Z/2Z. Let τ be a self-diffeomorphism over S 2 × S 1 with non- trivial homotopy class. The surgery X → X(S 2 × D 2 , τ ) is called a Gluck twist. 3.2. Proofs of Theorem 2. First we prove that Sr,s,o,k is diffeomorphic to S 4 . The deformation of the pictures in Figure 17 presents the Sr.s.o.k = S 4 . In the diagram of Sr.s.o.k the union of the (lowest) 0-framed 2-handle and a 4-handle consists of S 2 × D 2 . By removing the S 2 × D 2 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 × D 2 , τ m ) = S r,s,m,k . Thus S r,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. 3.3. The log transform along several tori. Here we define the log trans- form along n tori with the square zero. Let e i : T i 2 → X be disjoint em- bedded tori each other that the squares are all zero for i = 1, . . . n . Let c i be a curve presenting a primitive element in H 1 ( T 2 ). Let p i , q i be several pairs of coprime integers. Let é i be an embedding of the tubular neigh- borhood of T 2 with respect to e i . Suppose that a gluing diffeomorphism 9ci,pi,gi : T 2 × ∂D 2 → T 2 × ∂D 2 a diffeomorphism satisfying
∂D² → pi · ∂D² + qi · ci.
In fact, the image of ∂D 2 by the gluing map is the attaching sphere of the unique 2-handle in T 2 × D 2 and the remaining handles of T 2 × D 2 are two 3- handles and one 4-handle. The information of the image of ∂D 2 determines a diffeomorphism type.  Hence, the diffeomorphism type of the log transform along the tori depends only on e i , c i , p i and q i . We denote the log transform by
X[(eː), (cː), (pː/qː)].
and call ( p i /q i )-log transform along ( e i ) with direction ( c i ). When embed- dings ( e i ) and curves ( c i ) are clear in the context, we omit these items. 14
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3.4. Proof of Theorem 3. First, we find T 2 × D 2 in the diagrams in F IGURE 9 ( S r,s,m,1), in general, F IGURE 10 ( S r,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. One time (1/1)-log transform with direction ℓ corresponds to the change of the diagram which is given in Figure 13.  In general, the (1/s)-log transform is the s-times iteration of this process. Hence, S r,s,m,k is the (1/s)-log trans- 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 log transforms along two embedded disjoint tori in S 4 as in Figure 12. Namely, we have Sr,s,m,k = S 4 [(em,k,1,em,k,2),(c1,c2),(−1/r,1/s)] = S 4 [(−1/r,1/s)].
MOTOO TANGE
The two torus embeddings em,k,i : T i 2 → S 4 (i = 1,2) are embedded in such a way that each torus is embedded in each component T 2 × D 2 in b 2 T 2 × D 2 . The T 2 × D 2 b T 2 × D 2 exterior in S 4 is described in F IGURE 14. 3.5. A remark for the curves c 1 and c 2 . As mentioned in Section 1.6, if either of curves c 1 or c 2 in the boundary has an embedded slice disk in the exterior with framing –1. However, it is difficult to find such a disk in terms of the following observation. We consider a cobordism C from #²T² × S 1 to #³S 2 × S 1 by removing the three 3-handles and one 4-handle from the exterior which is described in Figure 14. We take an annulus in C beginning from either of c 1 or c 2 . Suppose that the annulus has no critical points in C , i.e., the annulus is the trace by the gradient flow of the Morse function for the handle decomposi- tion. Let č be the obtained knot in #³ 5 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 c has no −1-framed disk in h ³ D ³ × S ¹ . Because if there is such a disk, then by attaching a 2-handle on č with 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 ∞ can be regarded as 0-log transforms along the tori from the equality S r.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]. Soo,oo,m,k is obtained by exchanging two dots and two 0's in the sub- handle for k 2 T 2 × D 2 . The diagram of S ∞ ,∞,m,k is described in F IGURE 16. 16 By computing the fundamental groups and homology groups, the manifolds S∞,0,m,k and S∞,∞,m,k are homotopic to (S³ × S¹)#(S² × S²) and #²(S³ × S¹)#²(S² × S²) 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(S³ × S 1 )#²(S 2 × S 2 ). Thus essentially, we have to investigate the manifold S ∞ ,s 1,k and S ∞ ,∞ ,l,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. [1] D. Auckly, H. Kim, P. Melvin, and D. Ruberman, Equivariant corks, Algeb. Geom. Topol. 17 (2017) 1771–1783. [2] C. L. Curtis; M. H. Freedman; W. C.Hsiang; R. Stong, A decomposition theorem for h-cobordant smooth simply-connected compact 4-manifolds , Invent. Math. 123 (1996), no. 2, 343–348. [3] R. Fintushel and R. Stern, Knots, links and 4-manifolds , Invent. Math. 134 (1998), 363–400. [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, 152000, 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. 18 [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.jp