On Stein rational balls smoothly but not symplectically embedded in CP2$\mathbb {CP}^2$

We extend recent work of Brendan Owens by constructing a doubly infinite family of Stein rational homology balls which can be smoothly but not symplectically embedded in CP2$\mathbb {CP}^2$ .

Moreover, he showed that the pair ( 2 +1 , 2 −1 ) satisfies the Evans-Smith constraints only if = 1, and therefore that 2 +1 , 2 −1 does not embed symplectically in ℂℙ 2 for > 1. In this paper, we extend Owens' family of smooth embeddings to a two-parameter family of smooth embeddings , ⊂ ℂℙ 2 such that , cannot be symplectically embedded in ℂℙ 2 .
In [9], Owens also proves another result (Theorem 2), which states that a disjoint union of two or more of the balls ( ,1 ) cannot be smoothly embedded in ℂℙ 2 . This is viewed in [9] as mild support to a conjecture of Kollár [6], which would imply that at most three of the rational balls ( ,1 ) may embed smoothly and disjointly in ℂℙ 2 . It is therefore natural to ask whether the analogue of [9, Theorem 2] holds for our extended family or rational balls: Question 1. Can a disjoint union of two or more balls ( , ) be smoothly embedded in ℂℙ 2 ?
We plan to address Question 1 in a future paper. This paper is organized as follows. In Section 2, we fix notation and collect some preliminary material. Section 3 contains the proof of Theorem 1.

(ℤ)-framed chain links and (ℤ)-slam-dunks
Given a string of integers = ( 1 , … , ), let be a chain link consisting of oriented, framed unknots, with framing coefficients specified by . Performing Dehn surgery along each with coefficient gives rise, in the notation of Section 1, to the lens space ( ) = ( ). We shall need to keep track of detailed information about the gluing maps involved in the Dehn surgeries on the components of . In order to do that, we are going to view the framed link as an 2 (ℤ)-framed link in the sense of [5,Appendix], although we will use our own notation rather than the notation from [5]. Let ∶= 3 ⧵ be the complement of a tubular neighborhood ∶= 1 ⊔ … ⊔ of 1 ⊔ ⋯ ⊔ . We can express ( ) as the result of gluing solid tori 1 , … , to . The gluing maps ∶ → are determined up to isotopy by 2 × 2 matrices if we specify, for each of the tori and , two oriented curves that generate its first homology group -we identify such oriented curves with their homology classes and the maps with the induced maps in homology. We can do this as follows.
• Orient 1 , … , so that lk( , +1 ) = −1 ∀ . • In each , choose a canonical longitude with the same orientation as , and an oriented meridian that winds around according to the right-hand convention.
• Regarding each as the tubular neighborhood of an unknot in 3 , choose a canonical longitude and a meridian in as above. • For each , choose the basis ( , ) for 1 ( ) and the basis ( , ) for 1 ( ).
Note that, with these assumptions, and 1 , … , have compatible orientations if and only if the matrices representing 1 , … , with respect to the bases ( , ) and ( , ) have determinant 1. With this in mind, and recalling that each must be sent by −1 to + , we can choose with matrix , where denotes the matrix ∈ 2 (ℤ) and ∈ ℤ. (2.1) After these choices, each component is decorated with the matrix rather than simply with the integer , and becomes an 2 (ℤ)-framed link. Moreover, a presentation {( , )} =1 can be modified via 2 (ℤ)-slam-dunks (cf. [5, Lemma (A.2)]). We describe these modifications using our notation in the following proposition.
Proof. We first describe the case = 2. Let ′ be the lens space arising from Dehn surgeries along all the components of the chain link except 1 , so that ∶= ( ) is obtained from ′ by doing the remaining surgery along 1 . Since 1 is a meridian of 2 , we can isotope it, as an oriented knot in ′ , to − 2 = −1 2 ( 2 ) and then to the oriented core ′ 1 of 2 . See Figure 1, where the oriented curve representing −μ 2 , mapped by φ 2 to ℓ 2 , and the oriented curve representing λ 2 +a 2 μ 2 , mapped by φ 2 to m 2 , are shown. The isotopy from 1 to ′ 1 can be extended to an isotopy of tubular neighborhoods from 1 to ′ 1 ⊂ 2 , whose boundary is parallel to 2 . Now is obtained by cutting ′ 1 out of 2 and pasting 1 in its place, with the identification between ′ 1 and 1 given by a new gluing map ′ 1 . Note that, since 2 ⧵ ′ 1 is diffeomorphic to 2 × [0, 1], we may unambiguously take ( 2 , 2 ) as a basis for the domain of ′ 1 (regarded as a map between homology groups). With this assumption, ′ 1 is represented by the same matrix as 1 . In fact, as already observed, ′ 1 and 2 are isotopic as oriented knots, and ′ 1 admits 2 as a right-hand-oriented meridian. Hence, 2 and 2 also play the role of the original 1 and 1 . Moreover, it makes sense to consider the composition which is represented by the matrix 1 ⋅ 2 . This concludes the description of the 2 (ℤ)-slamdunk when = 2.
We now describe the construction for > 2 (assuming ⩾ 3). We first apply an 2 (ℤ)-slamdunk to the first component, so that 1 is removed from the chain link. Now 2 is a meridian of F I G U R E 1 A neighborhood of 2 3 , so we can apply another 2 (ℤ)-slam-dunk along 2 , and so on. In general, for each 1 ⩽ < , we remove a tubular neighborhood ′ of the core of +1 , and identify its boundary with via a gluing map ′ represented by with respect to the bases ( +1 , +1 ) and ( , ). By construction, for each = 2, … , the composition of gluing maps identifies = − −1 ( ) with a curve whose coordinates with respect to 1 and 1 are given by the second column of 1 ⋯ . Similarly, after gluing, the coordinates of with respect to 1 and 1 are given by the first column of 1 ⋯ . This proves (1) and (2). To prove (3), we choose = , so that the modified link has a single component. The result of gluing together all the 'layers' +1 ⧵ ′ for < is diffeomorphic to 2 × [0, 1] and the boundaries of the glued-up pieces are parallel tori. Moreover, the diffeomorphism with 2 × [0, 1] can be chosen so that:

PROOF OF THEOREM 1
The first part of Theorem 1 states that ( , ) smoothly embeds in ℂℙ 2 if is odd. We already observed in Section 1 that this is true if = −1, therefore in the following we assume ⩾ 0.
Consider the string , of Section 1, with ⩾ 0 and odd, and define: It is straightforward to check that the strings ′ , and ′′ , are both obtained from , by changing some terms from 2 to 1, and that they both 'blow-down' to (0) in the sense of [7, Definition 2.1], therefore ( ′ , ) = ( ′′ , ) = 1 × 2 .
We are going to prove Part (1) of Theorem 1 by showing that ℂℙ 2 is obtained by attaching some 4-dimensional handles to ( , ). First we attach two extra 2-handles along the meridians and ( +2) +2 , both with framing 1. Note that the indices and ( + 2) + 2 give the positions where , and ′′ , are different. As before, we rename these two meridians as 3 and 1 , respectively, so that we encounter 1 , 2 , and 3 in this order as we move along the diagram from right to left. Denote by the smooth 4-manifold with boundary constructed so far. If we view 1 , 2 and 3 as part of a surgery presentation and blow them down we get a chain of unknots whose framing coefficients are exactly given by ′′ , . This shows that the boundary of is 1 × 2 . We can now add a 3−handle and a 4−handle to and obtain a closed 4−manifoldˆ.
Our plan is to show thatˆis diffeomorphic to ℂℙ 2 . In order to do that, we view 1 , 2 , 3 ⊂ 1 × 2 as knots sitting inside a regular neighborhood of 1 ⊂ 1 × 2 as in Part (1) of Proposition 2.
The proof of Proposition 2 shows that can be identified with 2 × [0, 1] in such a way that each is identified with a simple closed curve 2 × {ℎ }, where 1 > ℎ 1 > ℎ 2 > ℎ 3 > 0. Moreover, the framing induced by on coincides with the framing induced by 2 × {ℎ }. We introduce the notation  to indicate that is -framed (with = ±1) with respect to the framing induced by 2 × {ℎ } and the coordinates of the homology class of with respect to the basis 1 , 1 are ( , ).
If we view 1 × 2 as ( ′ , ), applying Part (2) of Proposition 2 for = gives the standard presentation of 1 × 2 as ((0)), that is, as 0-surgery on an unknot. This way, 1 gets identified with the boundary of a neighborhood of such unknot, 1 with a longitude and 1 with a meridian.
Recall that, given a closed, oriented 3-manifold represented by a framed link with integer coefficients , there is a convenient way to represent handlebody decompositions of any 4-dimensional cobordism obtained by attaching 4-dimensional handles to × [0, 1] along × {1}. In fact, the attaching curves of the 2-handles can always be isotoped into the complement of the glued-in solid tori of × {1}, so that each 2-handle can be represented as an additional framed knot in 3 ⧵ . The union of all such framed knots with  is a relative Kirby diagram representing . This representation requires a notation which distinguishes the role played by each component. If the framing coefficient of a knot is , we are going to write it as ⟨ ⟩ if is part of , and simply as if represents a 2-handle of . Of course, we can also attach 3-and 4handles as usual. There is a calculus for these handlebody presentations, usually called relative Kirby calculus. We refer the reader to [2, § 5.5] for further details.
We are going to apply relative Kirby calculus to the handle decomposition ofˆwe just described. It turns out that the effect of sliding the handle ℎ attached along over (an appropriate number of copies of) the handle ℎ +1 attached over +1 , for = 1, 2, was described in [11,Lemma 5.1]. In terms of our Notation (3.1), the action of such handle slides on the triples of coordinates is given by the following sliding map , which can be applied to any two consecutive components of the triple as follows: Our strategy to prove thatˆis diffeomorphic to ℂℙ 2 will be as follows. We will exhibit a sequence of slides such that the coordinates and gradually get smaller, until we end up with a familiar Kirby diagram for ℂℙ 2 .
We now show that, for any pair ( , ) as above, the map transforms the starting triple ( 1 , 2 , 3 ) into In order to do that we need to determine the coordinates ( , ) of (3.1) in terms of and . These will be given by products of 2 × 2 matrices as in Proposition 2. Since the substring (2 [ −1] , + 2) occurs repeatedly in , , it will be useful to find a general formula for 2 ( −1 2 +2 ) (recall Notation 2.1). For this purpose, observe that an obvious induction gives  .
Lemma 5. The following hold.
By (3.4), both these equalities follow from 0 Δ 0 = − : this is true because Equivalence of handlebody decompositions used in the last step of the proof of Theorem 1 (1) where the second equality holds by Lemma 4(5). This proves (1). Finally, (2) and (3) follow from a straightforward computation; in particular, in order to prove (3), it is useful to observe that the quantity 0 Δ 0 stays unchanged at each step, since both 0 and Δ 0 change sign. □ Now, in order to prove Theorem 1(1), we must show that the triple 2 , corresponds to a Kirby diagram for ℂℙ 2 . By Lemma 5(1), it is enough to prove this for .
We can apply Lemma 5(2) several times to the last two components. Observe that all coordinates decrease by 2 at each step and recall that is odd. After −1 2 applications of Lemma 5(2), we get (( Proof. We have three knots in 2 × [0, 1] ⊂ 1 × 2 , which can be glued to 1 along 2 × {0} to form a new solid torus, which we regard as the exterior of an unknotˆin 3 (as in the proof of Proposition 2). Consequently, we can regard 1 × 2 as the result of a Dehn surgery alongˆwith framing 0. Now, the attaching curves 1 , 2 and 3 of the 2−handles are contained in three nested tori, each of which bounds a regular neighborhood ofˆ. More precisely, 1 is a parallel copy of 1 , hence a canonical longitude ofˆ, while 2 and 3 are two parallel copies of 1 , hence two unlinked meridians of bothˆand 1 . The left-most picture of Figure 2 illustrates the resulting handlebody decomposition. Performing the handle slide indicated by the horizontal arrow yields the second picture of Figure 2, canceling the obvious 1-2-handle pair yields the third picture, and canceling the 0-framed unknot with the 3-handle gives the well-known Kirby diagram for ℂℙ 2 . □ By Lemmas 5 and 6, the 4-manifoldˆis diffeomorphic to ℂℙ 2 . This proves the existence of the smooth embeddings, that is, Part (1) of Theorem 1.
The numbers above the equality symbols denote which identities from Lemma 4 have been used.

A C K N O W L E D G E M E N T S
The authors wish to thank Brendan Owens for a stimulating email exchange and an anonymous referee for helpful comments. The present work is part of the Ministero dell'Università e della Ricerca project 2017JZ2SW5. Open Access Funding provided by Universita degli Studi di Pisa within the CRUI-CARE Agreement.

J O U R N A L I N F O R M AT I O N
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