A cork of the rational surface with the second Betti number 9

We provide the first explicit example of a cork of $\mathbf{CP}^2 \# 8\overline{\mathbf{CP}^2}$. This result gives the current smallest second Betti number of a standard simply-connected closed $4$-manifold for which an explicit cork has been found.


Introduction
Throughout this article, we assume that all manifolds are smooth and oriented, and all maps are smooth unless otherwise stated.
One of the fascinating problems in 4-dimensional topology asks whether a simplyconnected closed 4-manifold with a small second Betti number b 2 admits an exotic smooth structure.The interest in this problem stems from the fact that constructing exotic smooth structures on such 4-manifolds is much more challenging than on those with large b 2 .The first example of an exotic smooth structure on simply-connected closed 4-manifolds was discovered by Donaldson [7,8].A few years later, Friedman and Morgan [11] proved that for each integer m ≥ 10, there exists a simply-connected closed 4-manifold with b 2 = m such that it admits infinitely many exotic smooth structures.After their result, many experts [20,25,27,26,3,4] have contributed to lowering the known minimal value of b 2 of a simply-connected closed 4-manifold on which an exotic smooth structure exists.Currently, we know that for each integer m ≥ 3, there exists a simply-connected closed 4-manifold with b 2 = m that admits infinitely many exotic smooth structures.
Regarding the exotic smooth structures on simply-connected closed 4-manifolds, the study of smooth structures of 4-manifolds by using the concept of cork (see Definition 3.1) plays an important role.Due to the work of Curtis-Freedman-Hsiang-Stong [6] and independently Matveyev [24], for any exotic pair (X, Y ) of simply-connected closed 4-manifolds, there exist a cork (C, τ ) and an embedding i of C into X such that Y is diffeomorphic to the cork twist of X along (C, τ, i), i.e., the 4-manifold obtained by cutting out the embedded copy i(C) in X and regluing C via the map i • τ .In other words, one can obtain any exotic smooth structure of X by a cork twist of X.When the cork twist of X along (C, τ, i) is exotic to X, we say i is an effective embedding of (C, τ ) into X.A cork (C, τ ) is called a cork of X if there exists an effective embedding of (C, τ ) into X.
Despite the importance of corks of simply-connected closed 4-manifolds, we have relatively few explicit examples of them.In particular, the minimal value of b 2 of standard simply-connected closed 4-manifolds for which an explicit cork had been found to date was 10 [1].A simply-connected closed 4-manifolds is called standard if it is obtained as the connected sum of finitely many copies of CP 2 , CP 2 , S 2 × S 2 , K3 and K3.As is well-known, if the celebrated 11/8-conjecture [23] is true, then it follows that any simplyconnected closed 4-manifold is homeomorphic to one of the standard simply-connected closed 4-manifolds.
This situation naturally leads us to the following problem, which can be regarded as a cork version of the problem we raised at the beginning of this paper.We remark that an effective embedding of the cork into a non-standard simply-connected closed 4-manifold with b 2 = 9 has already been found by Akbulut and Yasui [2, Remark 6.2], but it is unknown whether its cork twist results in a standard simply-connected closed 4-manifold.By the definition of cork twist, if the cork twist along their cork results in a standard simply-connected closed 4-manifold, it immediately follows that their cork has an effective embedding into a standard simply-connected closed 4-manifold.
In this article, we prove the following theorem by finding an effective embedding of a cork in Figure 1 into a standard 4-manifold.As a result, we provide an answer to Problem 1.1 in the case b 2 = 9.
Theorem 1.2.There exists an effective embedding of the cork (W 2 , f 2 ) into CP 2 #8CP 2 , i.e., the cork (W 2 , f 2 ) is a cork of CP 2 #8CP 2 .The proof of this theorem is divided into two parts.The first is giving an explicit Kirby diagram (Figure 19) of an exotic CP 2 #8CP 2 obtained by Yasui's construction [30,Corollary 5.2] which uses the rational blowdown technique [9].We denote this manifold as R 8 .Note that Yasui [30] explicitly described a procedure to give a Kirby diagram of a 4-manifold obtained by his construction.However, no explicit Kirby diagram of an exotic CP 2 #8CP 2 obtained by Yasui's construction was given before.We follow Yasui's procedure to give a diagram of R 8 with some modifications.The second is finding the diagram which contains an embedded copy of W 2 such that the cork twists along (W 2 , f 2 ) results in CP 2 #8CP 2 .Such a diagram is described in Figure 22.
The following follows from Theorem 1.2.
This corollary is related to the famous open problem asking whether every exotic pair of simply-connected closed 4-manifolds becomes diffeomorphic after one stabilization, i.e., taking a connected sum with S 2 × S 2 .It is well-known that, due to the theorem of Wall [29], every exotic pair of simply-connected closed 4-manifolds becomes diffeomorphic after sufficiently many stabilizations, and it has been proved that only one stabilization is enough in many cases.To the best of the author's knowledge, among the exotic pairs of simply-connected closed 4-manifolds whose Kirby diagrams are explicitly given, the pair (CP 2 #8CP 2 , R 8 ) is currently the smallest example in terms of b 2 that becomes diffeomorphic after one stabilization.It is not clear to the author whether other examples of exotic pairs of simply-connected closed 4-manifolds with b 2 ≤ 8 known to date become diffeomorphic after one stabilization.We note that many variants of the problem about one stabilization have recently been answered negatively.For details, see [21,22,13,19,16,14,15,17,18,5].
Remark 1.4.We can also prove an exotic CP 2 #kCP 2 (k = 5, 6, 7, 9) obtained by Yasui's construction becomes diffeomorphic to 2CP 2 #(k + 1)CP 2 after one stabilization.Furthermore, after posting the first version of this article on arXiv, Rafael Torres informed the author that a stronger version of Corollary 1.3 holds.Namely, the manifold R 8 becomes diffeomorphic to 2CP 2 #8CP 2 after taking a connected sum with CP 2 .It is possible to apply his idea to an exotic CP 2 #kCP 2 (k = 6, 7, 9) obtained by Yasui's construction.These results will be discussed in the forthcoming paper [28].
1.1.Acknowledgement.The author wishes to express his deepest gratitude to his advisor, Kouichi Yasui, for his patience, encouragement, and numerous valuable suggestions, including the topic of this paper.He is grateful to Rafael Torres for generously informing the author of a stronger version of Corollary 1.3 mentioned in Remark 1.4.He is thankful to the anonymous referees for their careful reading, many suggestions, and pointing out mistakes in the original manuscript.He also thanks Natsuya Takahashi for many valuable conversations and comments on the draft of this paper, and Yuichi Yamada for his interest in this study.

Rational blowdown and Yasui's small exotic rational surfaces
We start this section by reviewing the definition of the rational blowdown of 4-manifold, which was introduced by Fintushel and Stern [9].Then, we recall Yasui's construction [30] of an exotic CP 2 #8CP 2 that uses the rational blowdown.To find a cork embedded in an exotic CP 2 #8CP 2 in Section 3, we follow Yasui's construction [30] with small modifications (see Remark 2.6) and give an explicit diagram of the 4-manifold.Definition 2.1.For each integer p ≥ 2, let C p and B p be the compact 4-manifolds with boundary defined by the Kirby diagrams in Figure 2.Here u i in Figure 2 represents the elements of H 2 (C p ; Z) given by the corresponding 2-handles.(i.e., [12]).Let X be a compact 4-manifold and C be an embedded copy of C p in X.The 4-manifold X (p) = X − (intC) ∪ ∂ B p is called the rational blowdown of X along C.This operation is well-defined since any self-diffeomorphism of ∂B p extends over B p (For details, see [12,Section 8.5]).
Yasui's construction [30] begins with the following proposition.Although he constructed exotic CP 2 #kCP 2 for 5 ≤ k ≤ 9, we only focus on the case k = 8.Proposition 2.3 ([30, Proposition 3.1 (1)]).For a ≥ 1, the complex projective plane CP 2 admits the handle decomposition in Figure 3. Conventions 2.4.(1) In the figures below, we often draw only the local pictures of Kirby diagrams.We assume that the parts not drawn in the diagrams are naturally inherited from the previous diagrams and always fixed.
(2) In order to indicate the bands for the handle slides, we sometimes draw arrows in the diagrams as on the left in Figure 4. Figure 4 only shows the cases when the attaching circles of 2-handles are unknots and unlinked.As in Figure 4, the shape of the arrow determines the band for the handle slide.The arrow with a positive twist on the left in Figure 4 (b) will only appear in Figure 26   (3) We often represent the framings of the 2-handles by the second homology classes corresponding to the 2-handles.It is because we need the information of homology classes of certain 2-handles to prove Theorem 2.12.When the framings of the 2-handles are represented by the second homology classes, we mention the 2-handles or the attaching circles of 2-handles by their homology classes.Note that one can obtain the usual framing coefficients of the 2-handles by squaring their homology classes.(4) We denote the natural orthogonal basis of H 2 (CP 2 #kCP 2 ; Z) = H 2 (CP 2 ; Z) ⊕ k H 2 (CP 2 ; Z) by h, e 1 , e 2 , . . ., e k (i.e., h 2 = 1, e 2 i = −1, h • e i = 0, and e i • e j = 0(1 Proposition 2.5 (cf.[30, Proposition 3.2 (1), a = 4]).CP 2 #14CP 2 admits the handle decomposition in Figure 5.
Proof.The top left picture of Figure 6 shows the neighborhood of the full twist in Figure 3 for the case a = 4.By the procedure described in Figure 6, we obtain a Kirby diagram of CP 2 #2CP 2 shown in Figure 7.We isotope this diagram to obtain Figure 8. Then we can move one of the kinks on the left side of this diagram to obtain Figure 9.By another isotopy, we obtain Figure 10  Remark 2.6.The reader may notice that there are some differences between the Kirby diagrams in Figure 5 and [30, Proposition 3.2 (1), a = 4].First, we kept the attaching circles of all the 2-handles, whereas in [30] the 2-handles h, 2h, e 1 , e 2 , . . ., e 8 , and e 14 are omitted for simplicity.Second, we see that Figure 5 we reached is different from the picture as in [30, Figure 6, a = 4], even if we remove some attached 2-handles from     the other 2-handles from Figure 5, it looks like Figure 14.We can isotope this diagram to obtain Figure 15.We can find four positive full twists on the right side of Figure 15.
By an isotopy, we obtain Figure 16.We can find another four positive full twists from   Remark 2.9.Hereafter, we will not need to track the information of the homology classes of 2-handles.So, we will use framing coefficients to represent the framings of 2-handles in the following diagrams.We obtain Figure 18 by squaring the homology classes in Figure 5.
Now we will draw a diagram of R 8 .
Proof.Recall that there is a procedure to draw a diagram of a rational blowdown [2, Figure 16].If we apply this procedure to the copy of C 7 in Figure 5, we obtain Figure 19.
Remark 2.11.In [30, Proposition 3.9], Yasui showed that a 4-manifold obtained by his construction is homeomorphic to CP 2 #8CP 2 .To prove that it is homeomorphic to CP 2 #8CP 2 , he checked the simply-connectedness of his manifold to apply Rochlin's    theorem and Freedman's theorem.He also proved that a 4-manifold obtained by his construction is not diffeomorphic to CP 2 #8CP 2 by showing that its Seiberg-Witten invariant is non-trivial [30, Lemma 5.1 (1)].To prove the non-triviality of the Seiberg-Witten invariant, he first calculated a non-trivial value of the Seiberg-Witten invariant of CP 2 #14CP 2 .Then he used the information of the homology classes of the 2-handles of C 7 to apply the theorems of Fintushel and Stern [9] on the Seiberg-Witten invariant of the rational blowdown, and proved that a value of the Seiberg-Witten invariant of his manifold coincides with the above non-trivial value of the Seiberg-Witten invariant of CP 2 #14CP 2 .We followed Yasui's procedure [30] to construct C 7 in CP 2 #14CP 2 with small modifications.However, we can still apply the same arguments to prove that R 8 is an exotic  CP 2 #8CP 2 .First, by sliding a −1-framed unknot over the 0-framed unknot in the diagram in Figure 19, we obtain a diagram of R 8 where we can cancel the unique 1-handle.So R 8 is simply-connected, and one can check that R 8 is homeomorphic to CP 2 #8CP 2 by applying Rochlin's theorem and Freedman's theorem as in Yasui's argument.Also, the homology classes of the 2-handles of C 7 in Figure 5 and those in [30,Corollary 3.3 (1)] are the same.Therefore, the non-triviality of the Seiberg-Witten invariant of R 8 has also been proved by Yasui's argument.Thus, we obtain the following theorem.Theorem 2.12 (cf.[30, Corollary 5.2 (1)]).The 4-manifold R 8 is homeomorphic but not diffeomorphic to CP 2 #8CP 2 .

Finding a cork
In this section, we prove Theorem 1.2 and Corollary 1.3.To prove Theorem 1.2, we find an embedding of a cork into R 8 by using the diagram in Corollary 2.10, and show that the cork twist of R 8 along this cork results in the standard CP 2 #8CP 2 .First, let us briefly review the basic terminology associated with corks.Definition 3.1.Let (C, τ ) be a pair of a compact contractible 4-manifold with boundary and an involution τ on the boundary ∂C.We call (C, τ ) a cork if τ does not extend to any self-diffeomorphism on C. Let i be an embedding of C into a 4-manifold X.The 4-manifold X (C,τ,i) := (X − int(i(C))) ∪ i•τ C is called the cork twist of X along (C, τ, i).We say i is an effective embedding of (C, τ ) into X if X (C,τ,i) is not diffeomorphic to X.Note that for any cork (C, τ ) and any embedding i of C into X, the cork twist X (C,τ,i) is always homeomorphic to X by Freedman's result [10].A cork (C, τ ) is called a cork of a 4-manifold X if there exists an effective embedding of (C, τ ) into X.Lemma 3.2.(1) The 4-manifold R 8 admits the handle decomposition in Figure 22.
(2) The handle decomposition in Figure 22 contains a subhandlebody diffeomorphic to W 2 .
Proof.(1) Figure 20 (a) shows a local part of the diagram of R 8 in Figure 19.For the following argument, we often draw the local parts of the diagrams unless necessary.As mentioned in Section 2.4, the parts of the diagrams not drawn in the figures are always fixed.We slide −1-framed unknot over the 0-framed unknot, and then we isotope the diagram to obtain Figure 20 (b).We can isotope this diagram to Figure 20 (c).We slide the 0-framed unknot over the 6-framed unknot to obtain Figure 20 (d).By isotopies, we obtain Figure 20 (e), (f), and then Figure 21 (a).We slide the 4-framed unknot over the −1-framed unknot and then isotope the diagram to obtain Figure 21 (b).By another isotopy, we obtain Figure 21 (c).By sliding the 0-framed unknot in the diagram over one of the −1-unknots, we obtain Figure 21 (d).Now the entire diagram looks like Figure 22.Thus, R 8 admits the handle decomposition in Figure 22.
(2) The handle decomposition in Figure 22 contains the subhandlebody described in Figure 23.By isotoping this diagram likewise in Figure 14, 15, 16, and 17, we obtain Figure 24 (a).We isotope this diagram to obtain Figure 24 (b).By another isotopy, we can deform this diagram to Figure 24 (c), a diagram of W 2 .Therefore, the handle decomposition in Figure 22 contains a subhandlebody diffeomorphic to W 2 .Now we are ready to prove Theorem 1.2.
Proof of Theorem 1.2.From Lemma 3.2, we see that there exists an embedding of W 2 into R 8 .We call this embedding i.It is enough to show that the cork twist of R 8 along (W 2 , f 2 , i) results in CP 2 #8CP 2 .Figure 25 shows a diagram of the cork twist of R 8 along (W 2 , f 2 , i). Figure 26 (a) shows the bottom right part of this diagram.By following the steps described in Figure 26, we obtain Figure 26 (f).After performing a handle slide as shown in this figure, we can isotope the diagram to obtain Figure 27 (a).By following the steps described in Figure 27, we obtain Figure 27 (f).Note that the deformation from Figure 27 (b) to Figure 27 (c), in which the dot and 0 are exchanged, gives a diffeomorphism because the 0-framed unknot geometrically links with the dotted circle once, so this exchange represents cutting out a 4-ball and pasting it back.After a handle slide and isotopy, we obtain Figure 28 (a).By following the steps described in Figure 28, we obtain Figure 28 (f).After a handle slide and isotopy, we obtain Figure 29 (a).By following the steps described in Figure 29, we obtain Figure 29 (d).The whole Kirby diagram is shown in Figure 30, and we obtain Figure 31 by an isotopy.Figure 31 represents CP 2 #8CP 2 since we can obtain this diagram after performing 6 blowups in Figure 11 in the same manner we obtained Figure 12.
Proof of Corollary 1.3.First, recall that if M is a simply-connected 4-manifold, the result of the surgery along any embedded S 1 must be diffeomorphic to either M#S 2 × S 2 or M#S 2 ×S 2 ([12, Proposition 5.2.3]).Furthermore, if M is a non-spin simply-connected 4-manifold, then M#S 2 ×S 2 and M#S 2 ×S 2 are diffeomorphic ([12, Proposition 5.2.4]).Since the 4-manifolds R 8 and CP 2 #8CP 2 are non-spin and simply-connected, we only have to show that the results of the surgeries along embedded S 1 in those 4-manifolds are diffeomorphic.
By Theorem 1.2, we know that there exist diagrams of R 8 and CP 2 #8CP 2 such that each of diagrams contains a copy of the diagram of W 2 (in Figure 1), and becomes identical after changing a dotted circle into a 0-framed unknot.In general, changing a dotted circle into a 0-framed unknot corresponds to a surgery along an embedded S 1 [12, Section 5.4].

Problem 1 . 1 .
Find an effective embedding of a cork into a standard simply-connected closed 4-manifold with b 2 ≤ 9.

Figure 1 .
Figure 1.The cork (W 2 , f 2 ).The involution f 2 of ∂W 2 is defined by exchanging the zero and the dot in this diagram.
and Figure 28.As usual, the boxes with integer m in figures stand for m right-handed full twists if m is positive and |m| left-handed full twists if m is negative.

Figure 4 .
Figure 4.The arrows (on the left) determine the bands (on the right) for handle slides.

Figure 6 .
Figure 6.Isotopies and blowups applied to the diagram in a neighborhood of the full twist in Figure 3.

Figure 23 .
Figure 23.A subhandlebody contained in the handle decomposition in Figure22.