Constructions of homotopy 4-spheres by pochette surgery

The boundary sum of the product of a circle with a 3-ball and the product of a disk with a 2-sphere is called a pochette. Pochette surgery, which was discovered by Iwase and Matsumoto, is a generalization of Gluck surgery and a special case of torus surgery. For a pochette P embedded in a 4-manifold X, a pochette surgery on X is the operation of removing the interior of P and gluing P by a diffeomorphism of the boundary of P. We present an explicit diffeomorphism of the boundary of P for constructing a 4-manifold after any pochette surgery. We also describe a necessary and sufficient condition for some pochette surgeries on any simply-connected closed 4-manifold create a 4-manifold with the same homotopy type of the original 4-manifold. In this paper we construct infinitely many embeddings of a pochette into the 4-sphere and prove that homotopy 4-spheres obtained from surgeries along these embedded pochettes are all diffeomorphic to the 4-sphere by some explicit handle calculus and relative handle calculus.


Introduction
One of famous conjectures in 4-manifold topology is the 4-dimensional smooth Poincaré conjecture, which states that every homotopy 4-sphere is diffeomorphic to the 4-sphere.A Gluck surgery on a 4-manifold X is an operation of removing the interior of a tubular neighborhood of a 2-sphere in X with trivial normal Euler number from X and gluing D 2 ×S 2 by a non-trivial diffeomorphism of the boundary S 1 × S 2 .All Gluck surgeries on the 4-sphere create homotopy 4-spheres.Whether these manifolds are diffeomorphic to the 4-sphere is a well-known unsolved problem.The homotopy 4-sphere obtained by the Gluck surgery along a spun 2-knot or a 0-slice 2-knot is diffeomorphic to the 4-sphere [G, Me].
In 2004, Iwase and Matsumoto [IM] introduced a generalization of Gluck surgery called pochette surgery.A pochette is the boundary sum P = S 1 × D 3 ♮D 2 × S 2 of S 1 × D 3 and D 2 × S 2 .A pochette surgery on a 4-manifold X is an operation of removing the interior of P embedded in X from X and gluing P to X − int P by a diffeomorphism of ∂P .The diffeomorphism type of the manifold X ′ obtained by a pochette surgery along P embedded in X is determined by the embedding e : P → X, an element p/q of Q ∪ {∞} called the slope, and an element ε of {0, 1} called the mod 2 framing because the isotopy class of the gluing diffeomorphism of ∂P is characterized by p/q and ε.We denote X ′ by X(e, p/q, ε) and call it the pochette surgery on X for e, p/q, ε.The manifold X(e, 1/0, 1) is nothing but the Gluck surgery on X for the embedded 2-sphere e({0} × S 2 ).
Let DP = P ∪ (−P ) be the double of P and i P : P → DP the inclusion map.Kashiwagi [K] found an algorithm for drawing handle diagrams of pochette surgeries of DP and showed that DP (i P , 1/q, ε) is diffeomorphic to the Pao manifold L(q; 0, 1; ε)(see [P]).Murase [Mu] constructed handle diagrams of all pochette surgeries of DP and proved that DP (i P , p/q, ε) is diffeomorphic to L(q; 0, 1; ε).
We can consider P as h 0 ∪ h 1 ∪ h 2 , where h i is an i-handle for i = 0, 1, 2. Okawa [O] proved that if the pochette surgery S 4 (e, p/q, ε) is a homology 4-sphere and the core of e(h 1 ) is 'trivial' in S 4 − int e(h 0 ∪ h 2 ), then p must be 1.He also showed that S 4 (e, 1/q, ε) is diffeomorphic to the 4-sphere if e({0} × S 2 ) is a ribbon 2-knot.
In this paper we construct infinitely many embeddings of P into the 4-sphere and prove that homotopy 4-spheres obtained from surgeries along these embedded pochettes are all diffeomorphic to the 4-sphere.
The diagram depicted in Figure 1 is a handle diagram for the 4-sphere, where k is an integer greater than one, n = (n 1 , . . ., n k 2 −1 ) is a (k 2 − 1)-tuple of integers, and the sign of ±1 can be taken arbitrarily.Let e k,n : P → S 4 be the inclusion map from the pochette P which consists of the 0-handle, the 1-handle presented by the leftmost dotted circle, and the 2-handle presented by the rightmost 0-framed unknot in Figure 1.
Theorem 1.1.The pochette surgery S 4 (e k,n , 1/q, ε) on the 4-sphere S 4 for e k,n , 1/q, ε is diffeomorphic to S 4 for every k, n, q and ε (See also Figure 3).The diagram depicted in Figure 2 is also a handle diagram for the 4-sphere, where s, t are positive integers, m = (m 1 , . . ., m s ) is a s-tuple of integers, and n = (n 1 , . . ., n st+1 ) is a (st + 1)-tuple of integers such that s i=1 m i = 0. Let e m,n : P → S 4 be the inclusion map from P which consists of the 0-handle, the 1-handle presented by the leftmost dotted circle, and the 2-handle presented by the rightmost 0-framed unknot in Figure 2.
Theorem 1.2.The pochette surgery S 4 (e m,n , 1/q, ε) on the 4-sphere S 4 for e m,n , 1/q, ε is diffeomorphic to S 4 for every m, n, q and ε (See also Figure 4).In Section 2 we review a precise definition and known properties of pochette surgery.In Section 3 we give proofs of the main results.We assume that all manifolds are smooth, compact and oriented and all maps are smooth.
Acknowledgements.I am deeply grateful to my adviser, Hisaaki Endo for giving him courteous instructions in mathematics since I was a master's student.I would also like to express my sincere gratitude to Motoo Tange of the University of Tsukuba for contributing to his knowledge of pochette surgery and giving the definition of the mod 2 framings and methods of handle calculus.Finally I want to thank Koji Yamazaki for suggesting the relationship between pochette surgery on the 4-sphere and homotopy equivalence.Furthermore I want to thank them for teaching us how to write a math paper.

Preliminaries
Let X be a 4-manifold and E(A) the exterior X − int A of a subset A of X.Let Q e be the image e(Q) of a subset Q of P , e : P → X an embedding and g : ∂P → ∂E(P e ) a diffeomorphism.We call the curves l := S 1 × { * } and m := ∂D 2 × { * } on ∂P a longitude and a meridian of P , respectively.First, we define the pochette surgery.
Definition 2.1 (Iwase-Matsumoto [IM]).A pochette surgery on X is an operation of removing int P e and gluing in P by g : ∂P → ∂E(P e ).The 4-manifold E(P e )∪ g P obtained by the pochette surgery on X using e and g is denoted by X(e, g).The manifold X(e, g) is also called the pochette surgery on X for e and g.
In pochette surgery on a 4-manifold, after attaching D 2 × S 2 to P along g(m), the method of attaching S 1 ×D 3 is unique.Therefore, when gluing P , it is sufficient to consider an identification between neighborhoods of m and g(m) via g.
Fix an identification between ∂P and S 1 × ∂D 3 #∂D 2 × S 2 = S 1 × S 2 #S 1 × S 2 .The meridian m of P has the natural product framing.By embedding e, we get identification ι : ∂E(P e ) → S 1 × S 2 #S 1 × S 2 .Then, S 1 × S 2 #S 1 × S 2 can be expressed as the 2-component unlink which consists of 2 0-framed knots.Therefore, g maps the natural framing on m of ∂P to a framing on g(m).This framing on g(m) is represented by some integer determined by ι.The pochette can be regarded as S 1 × D 3 attaching a 2-handle with the cocore m.Let g 1 , g 2 : ∂P → ∂E(P e ) be two gluing maps.If g 1 (m) and g 2 (m) are the same and a difference between the framing on g 1 (m) and that of g 2 (m) is even, the map g −1 1 • g 2 | N (m) can be extended to the inside of the 2-handle.Here, N (A) is the open tubular neighborhood for a submanifold A of P .Therefore, when considering the diffeomorphism type of the pochette surgery, we should consider an integer modulo 2 as the framing on g(m).This framing on g(m) is called a mod 2 framing and write it as ε.The mod 2 framing of g(m) for the gluing map g : ∂P → ∂E(P e ) was first introduced in [IM,First paragraph in p.162].
By [IM,Lemma 4], the diffeomorphism type of X(e, g) is determined by the embedding e : P → X, the isotopy class of a simple closed curve g(m) and the mod 2 framing around g(m).For orientation preserving self-diffeomorphisms g, g [IM,Lemma 5]).Hence, the diffeomorphism type of X(e, g) is determined by an embedding e : P → X, a homology class g and the mod 2 framing around g(m).
Let p, q be coprime integers and ε an element of {0, 1}.By [IM, the seventh paragraph in p. 163], the homology class p The next theorem immediately follows from observations here (see [IM,Theorem 2]).
Let g p/q,ε : ∂P → ∂E(P e ) be a diffeomorphism which satisfies ] and the mod 2 framing of g p/q,ε (m) is ε in {0, 1}.We can define X(e, p/q, ε) = X(e, g p/q,ε ).From the construction, any pochette surgery for e, 1/0, ε is nothing but the Gluck surgery along S e , where S is the subset {0} × S 2 of P .
Suppose X is a homology 4-sphere, and i 11 : is the induced homomorphism of the composite map of the gluing map g : ∂P → ∂E(P e ) and the inclusion map i ∂E(Pe) : ∂E(P e ) ֒→ E(P e ).Okawa calculated some homology groups of the pochette surgery X(e, p/q, ε): Homology groups of pochette surgeries on any simply-connected closed 4-manifold will be calculated in Section 4.
Recall P can be interpreted as h 0 ∪ h 1 ∪ h 2 .We call the core of (h 1 ) e a cord.A cord is trivial if it is boundary parallel.Let S := { * } × S 2 ⊂ P .Okawa also showed that S 4 (e, 1/q, ε) is diffeomorphic to the 4-sphere if the cord (h 1 ) e is trivial in E((h 0 ∪ h 2 ) e ) and S e is a ribbon 2-knot.
Proof.This is a variation of [GS,Exercise 6.2.11(b)].

Handle diagram for pochette surgery
In this section we give a construction of handle diagrams for pochette surgeries under special conditions.Let X be a 4-manifold and e : P → X an embedding from a pochette P into X.Let p, q be coprime integers and ε an element of {0, 1}.Suppose that the diagram depicted in Figure 5 is a part of a handle diagram for X, where all the curves partially drawn in Figure 5 are framed knots, and any framed knot entwined with the dotted circle in Figure 5 has a 0-framed meridian.The pochette P e consists of the 0-handle, the 1-handle presented by the leftmost dotted circle, and the 2-handle presented by the rightmost 0-framed unknot in Figure 5.
Proposition 3.1.A handle diagram for X(e, p/q, ε) is depicted in Figure 6(A) if p/q is not equal to 0/1, and that for X(e, 0/1, ε) is depicted in Figure 6(B).
Proof.Here we will consider the case where only a framed knot is entwined with the 0-framed knot on the right side exactly once.The case where framed knots are entwined with the 0-framed knot on the right side can be proved in the same way.If |p| and |q| are coprime positive integers, then there exist a positive integer n, a nonnegative integer a 0 and positive integers a 1 , . . ., a n such that We define the diffeomorphism E 0 , E 1 , E 2 , E 3 , E 4 and E 5 : ∂P → ∂P as the 1-Rolfsen twist for the leftmost 0 -framed knot, the handle slide in Figure 7, 8, 9 and 10, the operation changing the direction of the meridian m, respectively.Then we have We define E p/q,ε to be and g p/q,ε = e • E p/q,ε .Then we have for any p/q ∈ Q ∪ {∞} and ε ∈ {0, 1}.By Theorem 2.2, the pochette surgery X(e, p/q, ε) is diffeomorphic to X(e, g p/q,ε ) for any p/q ∈ Q ∪ {∞} and ε ∈ {0, 1}.A part of a handle diagram of X is depicted in Figure 11.By several handle slides on the 0-framed meridians of the framed knots entwined with the dotted circle, we obtain a part of a handle diagram of X(e, p/q, ε) depicted in Figure 12.Concretely, the homotopy class of g p/q,ε (m) is the natural lift defined in [IM]: If pq = 0, then we reach the desired result.
If pq = 0, we obtain Figure 13 by creating a 2-handle/3-handle pair in Figure 12.By the handle slide in Figure 13 and several handle slides on the 0-framed meridians of the framed knots entwined with the dotted circle, we obtain the handle diagram depicted in Figure 14.By the handle slides between the leftmost 0-framed knot and the rightmost 0-framed knot in Figure 14: we obtain the handle diagram depicted in Figure 15.By the handle slide on a 0-framed meridian in Figure 15, the other 0-framed meridian in Figure 15 can be changed to 0-framed unlink.Changing the self-intersection of the framed knot in Figure 15 by several handle slides on the 0-framed meridian and canceling the 2-handle/3-handle pair, we obtain the handle diagram depicted in Figure 16.Therefore, we also obtain the conclusion in the case of pq = 0. Remark 3.2.If p = 1, Proposition 3.1 holds even without the 0-framed meridians, so any pochette surgery X(e, 1/q, ε) is given by Figure 6(A).

Homology of pochette surgery
Let p, q be coprime integers and ε an element of {0, 1}.Let B := { * } × ∂D 3 ⊂ P .Let X be a simply-connected closed 4-manifold and e : P → X an embedding from a pochette P into X.Here, we prove lemmas needed later.
Lemma 4.1.If the homomorphism t 2 : H 2 (X) → H 2 (X, E(P e )) induced by the inclusion map (X, ∅) → (X, E(P e )) is a zero map, the homology groups of the exterior E(P e ) of P e are calculated as follows: Here x is a point in ∂P .
Proof.By the long exact sequence of the pair (P, ∂P ): we have (n = 4), 0 (otherwise).By the Excision Theorem, we obtain H n (X, E(P e )) ∼ = H n (P, ∂P ) for any n ∈ Z.
By the long exact sequence of pair (X, E(P e )): we have the homology groups above.
Lemma 4.2.Let t n : H n (X) → H n (X, E(P e )), i n1 : H n (∂P ) → H n (E(P e )), i n2 : H n (∂P ) → H n (P ) be homomorphisms induced by inclusion maps.If Proof.By the definitions of E 0 , E 1 , E 2 , E 3 , E 4 and E 5 in the proof of Proposition 3.1, we obtain (i = 5) (double-sign corresponds).Then, there exist some integers r, s such that g p/q,ε * ( For a simply-connected closed 4-manifold X, we give a necessary condition for a pochette surgery of X to have the same homology as X. , then the homology groups of the pochette surgery X(e, p/q, ε) are calculated as follows: Moreover, if |p| is equal to 1, then X(e, p/q, ε) has the same homology groups as X.
Proof.Since X is connected and oriented, H n (X(e, p/q, ε)) ∼ = Z for any n = 0, 4. We compute H 1 (X(e, p/q, ε)) here.By Lemma 4.1 and the Mayer-Vietoris sequence we obtain the following: Then, we have i 11 Remark 4.4.Proposition 4.3 is a generalization of Theorem 2.3.
The next corollary follows from Proposition 4.3 and the Freedman theorem [F, FQ].
, then X(e, p/q, ε) is homeomorphic to X if and only if X(e, p/q, ε) is a simply connected 4-manifold and |p| is equal to 1.
Proof.If X(e, p/q, ε) is homeomorphic to X, then X(e, p/q, ε) has the same homology groups as X.By Proposition 4.3, X(e, p/q, ε) is a simply connected 4-manifold and |p| = 1.Conversely, if X(e, p/q, ε) is a simply connected 4-manifold and |p| = 1, we obtain a natural isomorphism H 2 (X(e, p/q, ε)) ∼ = H 2 (X) by the proof of Proposition 4.3.Hence, Q X(e,p/q,ε) ∼ = Q X .Here, Q Y is the intersection form of the 4-manifold Y .Since X(e, p/q, ε) and X are simply connected 4-dimensional closed manifolds with differential structures, X(e, p/q, ε) × R and X × R have differential structures.Therefore, we obtain ks(X(e, p/q, ε)) = 0 = ks(X).Here, ks(Y ) is the Kirby-Siebenmann invariant of Y .By the Freedman theorem, X(e, p/q, ε) is homeomorphic to X. Therefore, we obtain the desired result above.

Proofs of main theorems
Canceling the 1-handle/2-handle pairs and the 2-handle/3-handle pair in the handle diagrams depicted in Figures 1 and 2, we obtain the standard handle diagram of the 4-sphere which consists of a 0-handle and a 4-handle.Therefore both of the handle diagrams depicted in Figures 1 and 2 are those of the 4-sphere.By Proposition 3.1 and Remark 3.2, the handle diagrams depicted in Figures 3 and 4 are those of the manifold S 4 (e k,n , 1/q, ε) and the manifold S 4 (e m,n , 1/q, ε), respectively.Let H(n) be the union of n 3-handles and a 4-handle.
Proof of Theorem 1.1.We remove the 3-handle and the 4-handle in the handle diagram depicted in Figure 3. Taking the double of the obtained handle diagram and removing all the 1-handles and 2-handles that has existed since the handle diagram depicted in Figure 3, we obtain the handle diagram depicted in Figure 23.By several handle slides on 0-framed meridians and 0 -framed knots in Figure 23, 4k − 7 0-framed knots can be changed to 4k − 7 0-framed unknots.Canceling the 4k − 7 2-handle/3-handle pairs, we obtain the handle diagram depicted in Figure 24.By the handle slides in Figure 24, we obtain the handle diagram depicted in Figure 25.By several handle slides on the 2 0 -framed meridians in Figure 25, we obtain the handle diagram depicted in Figure 26.We can cancel the 2 Hopf links which consists of 2 • -framed knots in Figure 26.By the handle slides in Figure 26 and several handle slides on the 3 0 -framed meridians, we obtain the handle diagram depicted in Figure 27.By the handle slides in Figure 27, we obtain a Hopf link and a 0-framed meridian.By several handle slides on 0-framed meridians and 0 -framed knots in Figure 27, we obtain the handle diagram depicted in Figure 28.Repeating the handle calculus in Figure 24-28 in the same way, we obtain the handle diagram depicted in Figure 29.Here, L = k 2 −1 i=3 n i .By the handle slide in Figure 29, we obtain the handle diagram depicted in Figure 30.By several handle slides on the moved 0 -framed knot, we obtain the handle diagram depicted in Figure 31.Canceling the 2 2-handle/3-handle pairs and the 3 Hopf links, we obtain the handle diagrams depicted in Figure 32, 33 and 34 in order.Changing the 0 -framed knot in Figure 34 to a dotted circle, the upside down of the handle diagram depicted in Figure 3 is completed.Canceling the 1-handle/2-handle pair, we obtain the standard handle diagram of the 4-sphere which consists of a 0-handle and a 4-hanlde.
Proof of Theorem 1.2.By the handle slides in Figure 35 43) (double sign corresponds).Therefore, we have the handle diagram depicted in Figure 39 with q = 0 by the handle slide in Figure 40 or 42 |q| times.By canceling the s + 1 1-handle/2-handle pairs and the 2-handle/3-handle pair in the handle diagrams depicted in Figure 39 with q = 0, we obtain the standard handle diagram of the 4-sphere which consists of a 0-handle and a 4-handle.

Figure 1 .
Figure 1.A handle diagram of the 4-sphere.
, several handle slides on dotted circles and canceling the s(t − 1) 1-handle/2-handle pairs, we obtain the handle diagram depicted in Figure 36.Here, x i = t j=1 n s(j−1)+i for any i = 1, . . ., s.By the handle slides in Figure 36-37, we obtain the handle diagram depicted in Figure 38.Here, a, b 1 are some integers.Repeating the method of the handle calculus in Figure 37-38, we obtain the handle diagram depicted in Figure 39.Here, b 2 , . . ., b s−2 are some integers.By the handle slide in Figure 40 or 42, we obtain the handle diagram depicted in Figure 44.By the handle slide in Figure 44, we obtain the handle diagram depicted in Figure 45.By several handle slides on the 0-framed meridian, the handle diagram depicted in Figure 45 can be changed to that depicted in Figure 46.By the handle calculus in Figure 44-45 or 46 |m s | times, we obtain the handle diagram depicted in Figure 47.By the handle slide in Figure 47 and the method of the handle calculus in Figure 44-45 or 46 |m s−1 + m s | times, we obtain the handle diagram depicted in Figure 48.Here, c = m s−1 + m s .By repeating the method of the handle calculus in Figure 44-48 and s i=1 m i = 0, we can the (x 1 + ε)-framed knot in Figure 39 away from the dotted circles entwined with the n st+1 -framed knot.Thus, we obtain the handle diagram depicted in Figure 49.By the handle slide in Figure 40 (Figure 42) and the handle calculus in Figure 44-49, we obtain the handle diagram depicted in Figure 41 (Figure Figure 23