Orthogonal 2-Sphere Basis of Stable 4-Sphere

Abstract

Every stable 4-sphere is identified with the double branched covering space of a trivial surface-knot space. As a result of Wall, it is known that any two orthogonal bases of every stable 4-sphere are transformed into each other by an orientation-preserving diffeomorphism of the stable 4-sphere. In this paper another proof of Wall’s result is presented, and strengthened in the sense that the lift of an equivalence of the trivial surface-knot space can be taken as the diffeomorphism. Two applications are made. The first shows that every orientation-preserving diffeomorphism of every stable 4-sphere is nothing but the double branched covering lift of an equivalence of a trivial surface-knot space up to a smooth isotopy and a composition with an identity-shift. The second gives a similar result for TOP stable 4-spheres. Here, even if it is a smooth 4-manifold, unless it is diffeomorphic to the stable 4-sphere, the TOP trivial surface-knot space cannot be smooth.

1. Introduction

The stable 4-sphere of genus n is the connected sum Σ = Σ(n) of n copies of S² × S². The stable 4-sphere Σ(n) is canonically diffeomorphic to the double branched covering space S space S⁴(F)₂ of S⁴ branched along a trivial surface-knot F of genus n in S⁴; see [1]. Here, a trivial surface-knot of genus n in S⁴ is a surface-knot bounding a handlebody of genus n smoothly embedded in S⁴, whose pair (S⁴, F) is called a trivial surface-knot space of genus n. An orthogonal basis of the stable 4-sphere Σ(n) is a pairwise basis (x_∗, x′_∗) = {(x_i, x′_i)| i = 1, 2, ..., n} of the second integral homology group H₂(Σ(n); Z) which is a free abelian group of rank 2n such that the intersection numbers in Σ(n) have Int(x_i, x_j) = Int(x_i, x′_j) = Int(x′_i, x_j) = Int(x′_i, x′_j) = 0 for all i, j except for that Int(x_i, x′_i) = Int(x’_i, x_i) = 1 for all i. Known as Wall‘s result, any two orthogonal bases of every stable 4-sphere Σ are transformed into each other by an orientation-preserving diffeomorphism Σ of the stable 4-sphere Σ, [2]. The standard O2-sphere basis of the stable 4-sphere Σ(n) is the 2-sphere pair system (S² × 1_*, 1 × S²_*) = {(S²× 1_i, 1 × S²_i) | i = 1, 2, ..., n} in Σ(n). An O2-sphere basis of Σ(n) is a 2-sphere pair system (S_∗, S’_∗) = {(S_i, S′_i)| i = 1, 2, ..., n} of Σ(n), sent to the standard O2-sphere basis (S² × 1_*, 1 × S²_*) by an orientation-preserving diffeomorphism of Σ(n), or equivalently, which can rewrite the stable 4-sphere Σ(n) as the connected sum of the genus one stable 4-spheres S_i × S’_i (i = 1, 2, ..., n). Wall’s result is essentially equivalent to saying that every orthogonal basis (x_∗, x’_∗) of Σ(n) is represented by an O2-sphere basis (S_∗, S’_∗) of Σ(n), i.e., x_i = [S_i] and x’_i = [S’_i] (i = 1, 2, ..., n), because any two O2-sphere bases of Σ are transformed into each other by an orientation-preserving diffeomorphism of Σ(n).

In this paper, another proof of Wall’s result is presented, strengthened in the sense that the diffeomorphism σ is taken to be the lift f′ of an equivalence f of (S⁴, F) to the stable 4-sphere S⁴(F)₂ = Σ(n). Here, an equivalence of (S⁴, F) is an orientation-preserving diffeomorphism of S⁴ keeping the oriented trivial surface-knot F set-wise fixed. To state the main theorem (Theorem 1), the notion of an orthogonal 2-handle pair, or briefly an O2-handle pair on F, is needed. A 2-handle on a trivial surface-knot F is a 2-handle D × I on F smoothly embedded in S⁴ such that (D × I) ∩ F = (∂D) × I for a closed interval I with 0 as the center, and D × 0 is called the core disk of the 2-handle D × I and identified with D. An orthogonal 2-handle pair, or briefly an O2-handle pair on F in S⁴, is a pair (D × I, D′ × I) of 2-handle D × I and D′ × I on F which meet F only with the attaching annuli (∂D) ×I and (∂D′) ×I so that the loops ∂D and ∂D′ meet transversely at just one point x₀^, and the intersection (∂D) × I ∩ (∂D′) × I is diffeomorphic to the square Q = {x₀} ×I ×I; see Figure 1, [3]. For a trivial surface-knot F of genus n in S⁴, an O2-handle basis of F is a system (D∗ × I, D′∗× I) of mutually disjoint O2-handle pairs (Di × I, D′_i× I) (i = 1, 2, ..., n) on F in S⁴. Let p: Σ(n) = S⁴(F)₂ → S⁴ be the double branched covering projection branched along F, and α the nontrivial covering involution of S⁴(F)₂. The preimage p⁻¹(F) of F is the fixed-point set of α, diffeomorphic to F and written by the same notation as F. For every O2-handle basis (D∗ × I, D′∗× I) on F in S⁴, the O2-sphere basis (S(D_∗), S(D′_∗)) in Σ(n) is constructed so that S(D_i) = D_i ∪ αD_i and S(D′_i) = D′_i ∪ αD′_i (i = 1, 2, ..., n) are the preimages of the disks D_i and D′_i (i = 1, 2, ..., n) in S⁴ by the double branched covering projection p, respectively, [1]. The orientations of S(D_i) and S(D′_i) are taken with the orientations of D_i and D′_i and the opposite orientations of αD_i and αD′_i, respectively. The main theorem (Theorem 1) is stated as follows.

Theorem 1. 

Every stable 4-sphere Σ(n) is the double branched covering space S⁴(F)₂ of S⁴ branched along a trivial surface-knot F of genus n. Every orthogonal basis (x_∗, x′_∗) of Σ(n) is represented by an O2-sphere basis (S(E_∗^x), S(E′_∗^x)) of Σ(n) constructed from an O2-handle basis (E_∗^x × I, E′_∗^x × I) on F in S⁴. For any two orthogonal bases (x_∗, x′_∗) and (y_∗, y′_∗) of Σ(n), there are O2-handle bases (E_∗^x × I, E′_∗^x × I) and (E_∗^y × I, E′_∗^y × I) on F in S⁴ which are transformed into each other by an equivalence f of (S⁴, F) so that (x_∗, x′_∗) and (y_∗, y′_∗) are represented by the O2-sphere bases (S(E_∗^x), S(E′_∗^x)) and (S(E_∗^y), S(E′_∗^y)), respectively, which are transformed into each other by the lift f’ of f to S⁴(F)₂ = Σ(n).

A key to showing Theorem 1 is the following lemma.

Lemma 1 (Key Lemma). 

For every orthogonal basis (x_∗, x′_∗) of the stable 4-sphere Σ (n), there is an O2-handle basis (E_∗ × I, E′_∗ × I) on a trivial surface-knot F of genus n in S⁴ with Σ(n) =S⁴(F)₂ such that (x_∗, x′_∗) is represented by the O2-sphere basis (S(E_∗), S(E′_∗)) of Σ(n).

A loop basis of F is a pair system (e_∗, e′_∗) of oriented simple loop pairs (e_i, e′_i) (i = 1, 2, ..., n) on F which represents a basis for H₁(F; Z) such that e_i ∩ e_j = e′_I ∩ e′_j = e_i ∩ e′_j = ∅ for all distinct i, and j and e_i ∩ e′_i are one point with the intersection number Int(e_i, e′_i) = +1 in F for all i. Every oriented loop c on F bounds an immersed surface C in S⁴ with C ∩ F = c. The map q: H₁(F; Z) → Z₂ sending the homology class [c] in H₁(F; Z) to the Z₂-self-intersection number of C in S⁴ with respect to the F-framing is called the Z₂-quadratic function associated with the surface-knot F in S⁴. The identity q(x + y) = q(x) + q(y)+ Int_F(x, y)₂ for all x, y in H₁(F;Z) is used for calculation, where Int_F(x, y)₂ denotes the Z₂ -intersection number of x and y in F. A simple loop basis (e_∗, e′_∗) of F is spin if q(e_i) = q(e′_i) = 0 for all i. The loop pair system (∂D_∗, ∂D′_∗) for any O2-handle basis (D_∗ × I, D′_∗ × I) on F is a spin loop basis of F. Every handlebody smoothly embedded in S⁴ is smoothly isotopic to a standard handlebody in the equatorial 3-sphere S³ of S⁴. Thus, a trivial surface-knot F in S⁴ may be taken in the standard position in S³, where a standard O2-handle basis and a standard spin loop basis on F are taken. Any two spin loop bases on F are transformed into each other by an equivalence of (S⁴, F), [3] ((2.5.1), (2.5.2)), [4]. Hence every spin loop basis on F bounds the core disk pair system of an O2-handle basis on F. The core disk pair systems of any two O2-handle bases on F bounded by the same loop basis are transformed into each other by an equivalence of (S⁴, F), [5]. Under these preliminaries, the proof of Theorem 1 assuming Lemma 1 is done as follows.

Proof of Theorem 1 assuming Lemma 1. 

The first claim is shown, [1]. The second claim is shown by Lemma 1. If the third claim is proven, then the conclusion will be shown. By Lemma 1, let x_i = [S(E_i^x)], x′_i = [S(E′_i^x)] and y_i = [S(E_i^y)], y′_i = [S(E′_i^y)] for O2-handle bases (E_∗^x × I, E′_∗^x × I) and (E_∗^y × I, E′_∗^y × I) on a trivial surface-knot F of genus n in S⁴. Then there is an equivalence g₀ of (S⁴, F) sending the spin loop basis (∂E_∗^x, ∂E′_∗^x) of F to the spin loop basis (∂E_∗^y, ∂E′_∗^y) of F, [3,4]. For the O2-handle bases (g₀E_∗^x × I, g₀E′_∗^x × I) and (E_∗^y × I, E′_∗^y × I) on F with the same attaching part in F, there is an equivalence g of (S⁴, F) such that (gg₀E_∗^x × I, gg₀E′_∗^x × I) = (E_∗^y × I, E′_∗^y × I), [5]. The composite equivalence f = gg₀ of (S⁴, F) lifts to a diffeomorphism f′ of Σ (n)= S⁴(F)₂ sending the O2-sphere basis (S(E_∗^x), S(E′_∗^x)) to the O2-sphere basis (S(E_∗^y), S(E′_∗^y)) and hence sending (x_i, x′_i) = ([S(E_i^x)], [S(E′_i^x)]) to (y_i, y′_i) = ([S(E_i^y)], [S(E′_i^y)]) for all i. This completes the proof of Theorem 1 assuming Lemma 1. □

Theorem 1 is a revised version of an incorrect claim [6] (Lemma 3.1) that was originally intended for use in the paper [1], which was written instead by using Wall’s result [2]. The proof of Key Lemma (Lemma 2) is done in Section 2. Two applications of Theorem 1 are done in Section 3 and Section 4. In Section 3, it is shown that every orientation-preserving diffeomorphism of Σ is nothing but the lift of an equivalence of a trivial surface-knot space (S⁴, F) to the double branched covering space Σ = S⁴(F)₂ up to a smooth isotopy and a composition of an identity-shift. In Section 4, a TOP version of Theorem 1 is done for every TOP stable 4-sphere of genus n (i.e., topological 4-manifold homeomorphic to the stable 4-space of genus n). Here, even if it is a smooth 4-manifold, unless it is diffeomorphic to the stable 4-sphere of genus n, the TOP trivial surface-knot space cannot be smooth.

2. Proof of Key Lemma (Lemma 1)

For a disk D, let D^o = D \ ∂D. The following lemma gives basic information on the intersection numbers of the lifting O2-sphere bases of two O2-handle bases of F in S⁴, which corrects a computation error of [6] (Lemma 3.1).

Lemma 2. 

For an O2-handle basis (D_∗× I, D′_∗× I) on a trivial surface-knot F of genus n in S⁴, let (k_∗, k′_∗) = (∂D_∗, ∂D′_∗) be the spin loop basis of F. For a 2-handle E × I on F in S⁴, assume that the homology class [e] of the simple loop e = ∂E in F is given by the intersection numbers Int([e],[k’_j]) = s_j and Int([e],[k_j]) = s’_j in F for some integers s_j, s′_j (j = 1, 2, ..., n). Then the homology class [S(E)] in Σ is written as [S(E)] = Σⁿ_{j = 1} (s_j + 2m_j) [S(D_j)] + Σⁿ_{j = 1} (s′_j + 2m′_j) [S(D′_j)], where m_j and m′_j are integers given by the intersection numbers Int (E^o, D′_j^o) and Int (E^o, D_j^o) in S⁴, respectively.

Proof of Lemma 2. 

Let N(F)^c = cl(S⁴ \N(F)) for a regular neighborhood N(F) of F in S⁴. Consider that the disk E^e = E ∩ N(F)^c transversely meets the disks D_j^e = D_j ∩ N(F)^c and D′_j^e = D′_j ∩ N(F)^c with the intersection points E^o ∩ D_j^o and E^o ∩ D′_j^o for all i, respectively. The intersection number of the lift of E^e and the lift of D_j^e to S⁴(F)₂ = Σ is equal to Int(E^e, D_j^e) + Int(αE^e, αD_j^e) = 2 Int(E^e, D_j^e) = 2 Int(E^o, D_j^o) = 2m′_j. Similarly, the intersection number between the lift of E^e and the lift of D′_j^e to S⁴(F)₂ = Σ is equal to Int(E^e,D′_j^e) + Int(αE^e, αD′_j^e) = 2 Int(E^e, D′_j^e) = 2 Int(E^o, D′_j^o) = 2m_j. By using the intersection numbers Int([e],[k_j]) = s’_j, Int([e],[k’_j]) = s_j in F and examining the geometric intersections between the lift of E ∩ N(F) and the lifts of D_j ∩ N(F), D′_j ∩ N(F) to S⁴(F)₂ = Σ, the identities Int([S(E)],[S(D_j)]) = s′_j + 2m′_j, Int([S(E)],[S(D′_j)]) = s_j + 2m_j are obtained. Since ([S(D_*)],[S(D′_*)]) is an orthogonal basis of Σ, the desired identity is obtained. This completes the proof of Lemma 2. □

By using Lemma 2, the following lemma is obtained.

Lemma 3. 

For every orthogonal basis (x_∗, x′_∗) of Σ, there is an O2-handle basis (E_∗× I, E′_∗× I) on the trivial surface-knot F of genus n in S⁴ such that x_i = [S(E_i)] + 2A_i + 2A′_i, x′_i = [S(E′_i)] + 2B_i + 2B′_i for all i, where A_i= Σⁿ_{j = 1} a_ij[S(E_j)], A′_i= Σⁿ_{j = 1} a′_ij[S(E′_j)], B_i= Σⁿ_{j = 1} b_ij[S(E_j)], and B′_i = Σⁿ_{j = 1} b′_ij[S(E′_j)] with some integers a_ij, a′_ij, b_ij, b′_ij for all i, j.

Proof of Lemma 3. 

For an O2-handle basis (D_∗ × I, D′_∗× I) on a trivial surface-knot F of genus n in S⁴, write x_i and x′_i as integral linear combinations of the homology classes on [S(D∗)] and [S(D′∗)] such that x_i = Σⁿ_{j = 1} c_ij [S(D_j)] + Σⁿ_{j = 1} c′_ij [S(D′_j)], x′_i = Σⁿ_{j = 1} d_ij [S(D_j)] + Σⁿ_{j = 1} d′_ij [S(D′_j)] for all i. Since (x_∗, x′_∗) and ([S(D_*)],[S(D′_*)]) are orthogonal bases of Σ, the identities Σⁿ_{j = 1} c_ijc′_ij = Σⁿ_{j = 1} d_ijd′_ij = 0, Σⁿ_{j = 1} (c_ijd′_ij + c′_ijd_ij) = 1 hold for all i. Let (k_∗, k′_∗) = (∂D_∗, ∂D′_∗) be a spin loop basis of F. Then there is a spin loop basis (ℓ_∗, ℓ′_∗) on F such that [ℓ_i] = Σⁿ_{j = 1} c_ij [k_j] + Σⁿ_{j = 1} c′_ij [k′_j], [ℓ′_i] = −Σⁿ_{j = 1} d_ij [k_j] + Σⁿ_{j = 1} d′_ij [k′_j] for all i in H₁(F; Z). In fact, a simple loop basis (ℓ_*, ℓ′_*) on F with the identities above is constructed by a diffeomorphism realization between symplectic bases of F. Then the Z₂-quadratic function q: H₁(F; Z) → Z₂ gives q([ℓ_i]) =Σⁿ_{j = 1} c_ijc′_ij = 0, q([ℓ′_i]) = Σⁿ_{j = 1} d_ijd′_ij = 0 for all i, showing that the simple loop basis (ℓ_*, ℓ′_*) is a spin loop basis on F. Let (E_* ×I, E′_*× I) be an O2-handle basis of F with (∂E_*, ∂E′_*) = (ℓ_*.ℓ′_*). By Lemma 2, [S(E_i)] = Σⁿ_{j = 1}(c_ij + 2m_ij) [S(D_j)] + Σⁿ_{j = 1}(c′_ij + 2m′_ij) [S(D′_j)], [S(E′_i)] = Σⁿ_{j = 1}(d_ij + 2n_ij) [S(D_j)] + Σⁿ_{j = 1}(d′_ij + 2n′_ij) [S(D′_j)] with some integers m_ij, m′_ij, n_ij, n′_ij for all i, j, Thus, x_i = [S(E_i)] − 2Σⁿ_{j = 1}m_ij[S(D_j)] − 2Σⁿ_{j = 1}m′_ij[S(D′j)], x′_i = [S(E′_i)] − 2Σⁿ_{j = 1}n_ij[S(D_j)] − 2Σⁿ_{j = 1}n′_ij [S(D′j)] for all i. The homology classes [S(D_j)], [S(D′_j)] are Z-linear combinations of the basis ([S(E_*)], [S(E′_*)]) of H₂(Σ;Z), the desired identities with some A_i, A′_i, B_i, B′_i are obtained. This completes the proof of Lemma 3. □

Let S⁴ be the one-point compactification of the 4-space R⁴. For the 3-space R³ and an interval J ⊂ R, the notation R³J = { (x, t) ∈ R⁴| x ∈ R³, t ∈ J } is used. Consider the trivial surface-knot F as a standard surface in R³ and the O2-handle basis (D_∗ × I, D′_∗ × I) on F is embedded in R³. Let D_i⁺ be a slightly extended disk of D_i so that the boundary loop ∂(D_i⁺) is disjoint from F and meets the disk D′_i transversely at a single point, and D′_i⁺ a slightly extended disk of D′_i so that the boundary loop ∂(D′_i⁺) is disjoint from F and meets the disk D_i transversely at a single point; Figure 2. A 2-sphere surrounding D_i is a 2-sphere S_i in R⁴ which is the boundary of the 3-ball D_i⁺[−2,1] in R⁴, and a 2-sphere surrounding D′_i is a 2-sphere S′_i in R⁴ which is the boundary of the 3-ball (D′_i)⁺[−1,2] in S⁴. The 2-spheres S_i, S′_i (i = 1, 2, ..., n) are disjoint for distinct indexes i and meet transversely in just two points with opposite signs in R³[±1] for the same index i. The 2-sphere pair (S_i, S′_i) in S⁴ is called Montesinos’s twin, Figure 3, [7]. The connected sum D_i # S_i is made along an arc β′ in S′_i joining the intersection point D_i ∩ S′_i in R³[0] with the intersection point of S_i ∩ S′_i in R³, [1]. The 2-sphere S_i is oriented so that the disk D_i # S_i is oriented with the orientation inherited from D_i. Similarly, the connected sum D′_i # S′_i is made along an arc β in S_i joining the intersection point D′_i ∩ S_i in R³[0] with the intersection point of S_i ∩ S ′_i in R³[−1]. The 2-sphere S′_i is oriented so that the disk D′_i # S′_i is oriented with the orientation inherited from D′_i. To show Lemma 1, the following lemma is used.

Lemma 4. 

The homology classes [S_i] and [S′_i] in Σ are given by [S_i] = −[S(D_i)], [S′_i] = −[S(D′_i)] for all i.

Proof of Lemma 4. 

From construction, S_i and S′_i are disjoint from S(D_j) and S(D’_j) for every j ≠ i, and the intersection numbers Int([S_i],[S(D_i)]) = Int([S_i],[S_i]) = Int([S’_i],[S′_i]) = Int([S’_i],[S(D’_i)]) = 0, and Int([S_i], [S(D’_i)]) = ε and Int([S′_i],[S(D_i)]) = ε′ for ε, ε′ = ±1, so that [S_i] = ε[S(D_i)], [S′_i] = ε′[S(D′_i)]. By the identities [αS(D_i)] = −[S(D_i)], Int([αS(D_i)],[αS′_i]) = Int([S(D_i)],[S′_i]), the identity [αS′_i] = −[S′_i] is obtained. Similarly, since [αS(D′_i)] = −[S(D′_i)], Int([αS(D′_i)],[αS_i]) = Int([S(D′_i)], [S_i]), the identity [αS_i] = − [S_i] is obtained. Let (E_∗ × I, E′_∗ × I) be the O2-handle basis on F given by (E_i, E′_i) = (D_i#S_i, D′_i #S′_i) and (E_j, E′_j) = (D_j, D′_j) for j ≠ i. Then the identities [S(E_i)] = [S(D_i] + [S_i] − [αS_i] = [S(D_i)] + 2[S_i], [S(E′_i)] = [S(D′_i] + [S′_i] − [αS′_i] = [S(D′_i)] + 2[S′_i] are obtained in Σ. The desired identities [S_i] = − [S(D_i)], [S′_i] = − [S(D′_i)] are obtained from the identities [S_i] = ε[S(D_i)], [S′_i] = ε′ [S(D′_i)] and Int([S(E_i)],[S(E′_i)]) = Int([S(D_i],[S(D′_i)]) = 1. This completes the proof of Lemma 4. □

As a by-product of the proof of Lemma 4, an elementary transformation on an O2-handle basis on a trivial surface-knot in S⁴ was found to change an O2-sphere basis of Σ. In fact, the following elementary transformation on the O2-handle basis (D_∗ × I, D′_∗ × I) on F is given by Lemma 4.

Operation 1. 

An O2-handle basis (D_∗ × I,D′_∗ × I) on F is changed into an O2-handle basis (E_∗× I, E′_∗× I) on F with the same attachment as (D_∗ × I, D′_∗ × I) such that [S(E_i)] = −[S(D_i)], [S(E′_i)] = −[S(D′_i)] in Σ and (E_j, E′_j) = (D_j, D′_j), j ≠ i, for any given i.

Here are further elementary transformations on the O2-handle basis (D_∗ × I, D′_∗ ×I) on F.

Operation 2. 

Let i ≠ j and join the intersection point v_i = D_i ∩ S′_i with the intersection point v′_j = D′_j ∩ S_j by an arc γ in S⁴ whose interior is disjoint from F and (D_* × I, D′_* × I), where the 2-spheres S′_i and S_j are taken to be unoriented. Construct the connected sum pairs (E_i, E′_i) = (D_i # S_j, D′_i) and (E_j, E′_j) = (D_j, D′_j # S′_i) along the arc γ. For h ≠ i, j, let (E_h, E′_h) = (D_h, D′_h). Then (E_* × I, E′_* × I) is an O2-handle basis on F with the same attachment as (D_* × I, D′_* × I) such that [S(E_i)] = [S(D_i)] −2ε [S(D_j)], [S(E′j)] = [S(D′_j)] +2ε [S(D′_i)] with ε = ±1 in Σ and E′_i = D′_i, E_j = D_j, (E_h, E′_h) = (D_h,D′_h), h ≠ i, j for any given distinct indexes i and j.

In fact, the identities above are obtained because in the following identities [S(E_i)] = [S(D_i] + 2ε[S_j] = [S(D_i)] − 2ε[S(D_j)], [S(E′_j)]) = [S(D′_j)] + 2ε′[S′_i] = [S(D′_j)] − 2ε′[S(D′_i)] with ε′ = ±1 given by Lemma 4, the identity ε + ε′ = 0 is obtained from the intersection number Int([S(E_i)], [S(E′_j)]) = 0.

Operation 3. 

Let i ≠ j and join the intersection point v_i = D_i ∩ S′_i with the intersection point v′_j = D_j ∩ S′_j by an arc γ in S⁴ whose interior is disjoint from F and (D_* × I, D′_* × I), where the 2-spheres S′_i and S′_j are taken to be unoriented. Construct the connected sum pairs (E_i, E′_i) = (D_i # S′_j, D′_i) and (E_j, E′_j) = (D_j # S′_i, D′_j) along the arc γ. For h ≠ i, j, let (E_h, E′_h) = (D_h, D′_h). Then (E_* × I, E′_* ×I) is an O2-handle basis on F with the same attachment as (D_* × I, D′_* × I) such that [S(E_i)] = [S(D_i)] − 2ε [S(D′_j)], [S(E_j)] = [S(D_j)] + 2ε [S(D′_i)] with ε = ±1 in Σ and E′_i = D′_i, E′_j = D′_j, (E_h, E′_h) = (D_h, D′_h), h ≠ i, j for any given distinct indexes i and j.

In fact, the identities above are obtained because in the following identities [S(E_i)] = [S(D_i)] + 2ε[S′_j] = [S(D_i)] − 2ε[S(D′_j)], [S(E_j)]) = [S(D_j)] + 2ε′[S_i] = [S(D_j)] − 2ε′[S(D′_i)] with ε′ = ±1 given by Lemma 4, the identity ε + ε′ = 0 is obtained from the intersection number Int([S(E_i)], [S(E_j)]) = 0.

Operation 4. 

Let i ≠ j and join the intersection point v_i = D′_i ∩ S_i with the intersection point v′_j = D′_j ∩ S_j by an arc γ in S⁴ whose interior is disjoint from F and (D_* × I, D′_* × I), where the 2-spheres S_i and S_j are taken to be unoriented. Construct the connected sum pairs (E_i, E′_i) = (D_i, D′_i # S_j) and (E_j, E′_j) = (D_j, D′_j # S_i) along the arc γ. For h ≠ i, j, let (E_h,E′_h) = (D_h, D′_h). Then (E_* × I, E′_* × I) is an O2-handle basis on F with the same attachment as (D_* × I, D′_* × I) such that [S(E′_i)] = [S(D′_i)] −2ε [S(D_j)], [S(E′_j)] = [S(D′_j)] +2ε [S(D_i)] with ε = ±1 in Σ and E_i = D_i, E_j = D_j, (E_h, E′_h) = (D_h, D′_h), h ≠ i, j for any given distinct indexes i and j.

In fact, the identities above are obtained because in the following identities [S(E′_i)] = [S(D′_i)]+ 2ε[S_j] = [S(D′_i)] − 2ε[S(D_j)], [S(E′_j)]) = [S(D′_j)] + 2ε′[S_i] = [S(D′_j)] − 2ε′[S(D_i)] with ε′ = ±1 given by Lemma 4, the identity ε + ε′ = 0 is obtained from the intersection number Int([S(E′_i)], [S(E′_j)]) = 0.

In Operations 2–4, the sign ε can take both +1 and −1, because a normal 3-disk bundle of the arc γ in S⁴ admits a Hopf-link bundle of the arc γ which is used for the connected sum and the Hopf link has a component-preserving inversion. The following observation on an odd integer u and a non-zero integer c is used for the proof of Lemma 2.

Observation 1. 

Let u be an odd integer, c a non-zero integer, and ε = ±1 the sign of the product integer uc. If |c| ≥ |u|, then 0 ≤ |2c − 2εu| < 2|c|. If |u| > |c|, then the integer u − 2εc is odd with 1 ≤ |u − 2εc| < |u|.

In fact, let ε′c > 0 for ε′ = ±1. If ε′c ≥ ε′εu > 0, then 0 ≤ 2ε′c − 2ε′εu < 2ε′c, showing that |c| ≥ |u| implies 0 ≤ |2c − 2εu| < 2|c|. If ε′εu > 2ε′c > 0, then 0 < ε′εu −2ε′c < ε′εu. If 2ε′c > ε′εu > ε′c > 0, then 0 > ε′εu −2ε′c > −ε′c > −ε′εu. These inequalities imply 1 ≤ |u − 2εc| < |u|, as desired.

In applications of Observation 1 to the proof of Lemma 1, it is shown that the coefficients u_i = 1 + 2a_ii of [S(E_i)] in x_i and u’_i = 1 + 2a’_ii of [S(E’_i)] in x′_i are odd integers. The proof of Lemma 1 is done as follows.

Proof of Lemma 1. 

Fix the orthogonal basis (x_*, x′_∗) of Σ. By Lemma 3, there is an O2-handle basis (E_* × I, E′_* × I) on F such that x_i = [S(E_i)] + 2A_i + 2A′_i, x′_i = [S(E′_i)] + 2B_i + 2B′_i for all i, where A_i = Σⁿ_{j = 1} a_ij[S(E_j)], A′_i = Σⁿ_{j = 1} a′_ij[S(E′_j)], B_i = Σⁿ_{j = 1} b_ij[S(E_j)], B′_i = Σⁿ_{j = 1} b′_ij[S(E′_j)] with some integers a_ij, a′_ij, b_ij, b′_ij for all i, j. Let n = 1. Since (x₁, x′₁) and ([S(E₁)], [S(E′₁)]) are orthogonal bases of Σ, it holds that a′₁₁ = b′₁₁ = 0 and either a₁₁ = b₁₁ = 0 or a₁₁ = b₁₁ = −1, so that either x_i = [S(E_i)], x′_i = [S(E′_i)] or x_i = −[S(E_i)], x′_i = −[S(E′_i)]. For the latter case, apply Operation 1 to obtain x₁ = [S(E₁)] and x′₁ = [S(E′₁)]. Let n ≥ 2. By a finite number of Operation 2 and Observation 1 using the transformations [S(E_n)] → [S(E_n)] ± 2[S(E_j)] and [S(E_j)] → [S(E_j)] ± 2[S(E_n)] for j < n, there is an O2-handle basis (E_∗ × I, E′_∗ × I) on F such that x_n = ε[S(E_n)] + 2A′_n(ε = ±1), x′_n = [S(E′_n)] + 2B_n + 2B′_n for some A’_n, B_n, B’_n. By a finite number of Operation 3 and Observation 1 using the transformations [S(E_n)] → [S(E_n)] ± 2[S(E′_j)] for j < n, there is an Identity x_n = ε[S(E_n)] + 2a’_nn[S(E′_n)] for some integer a’_nn. Then a’_nn = 0, because Int(x_n, x_n) = Int([S(E_n)], [S(E_n)]) = Int([S(E’_n)], [S(E’_n)]) = 0 and Int}([S(E_n)], [S(E’_n)]) = 1. Thus, x_n = ε [S(E_n)], x′_n = [S(E′_n)] + 2B_n + 2B′_n with some B_n, B′_n. By a finite number of Operation 4 and Observation 1 using the transformations [S(E′_n)] → [S(E′_n)] ± 2[S(E_j)] for j <n, the identities x_n = ε [S(E_n)], x′_n = [S(E′_n)] + 2b_nn[S(E_n)] + 2B′_n are obtained for some integer b_nn and some B′_n. By Operation 1, assume that ε =1. Use that (x_∗, x′_∗) and ([S(E_∗)], [S(E′_∗)]) are orthogonal bases of Σ. Since Int(x′_n, x′_n) = 4b_nn = 0 and thus b_nn = 0, the identities x_n = [S(E_n)], x′_n = [S(E′_n)] + 2B′_n are obtained. Then B′_n = 0, for Int(x_j, x′_n) = 0 (j < n) or 1(j = n), and the identities x_n = [S(E_n)], x′_n = [S(E′_n)] are obtained. Let F_n−1 be the trivial surface-knot of genus n−1 in S⁴ obtained from F by surgery along the O2-handle pair (E_n × I, E′_n × I), [3]. Then the system (x_i, x′_i) (i = 1, 2, ..., n − 1) is an orthogonal basis for the stable 4-sphere Σ(n−1) = S⁴(F_n−1)₂ of genus n−1. By inductive assumption on n, there is a replacement of the O2-handle basis (E_i× I,E′_i × I) (i = 1, 2, ..., n − 1) on F_n−1 keeping the attaching part fixed so that x_i = [S(E_i)] and x_i = [S(E′_i)] for all i (i = 1, 2, ..., n − 1) in Σ(n − 1). Thus, there is an O2-handle basis (E_∗ × I, E′_∗× I) on F such that x_i = [S(E_i)] and x′_i = [S(E′_i)] for all i (i = 1, 2, ..., n) in Σ = Σ(n), [3]. This completes the proof of Lemma 1. □

This completes the proof of Theorem 1.

3. Application to Diffeomorphism of Stable 4-Sphere

The orientation-preserving diffeomorphism group of the 4-ball D⁴ keeping the boundary ∂D⁴ point-wise fixed is denoted by Diff⁺(D⁴, rel∂). An identity-shift in the stable 4-sphere Σ is a diffeomorphism ι : Σ → Σ obtained from the identity 1 : Σ → Σ by replacing the identity on a 4-ball in Σ = S⁴(F)₂ disjoint from F with an element of Diff⁺(D⁴, rel∂). The following theorem is shown as an application of Theorem 1.

Theorem 2. 

For every stable 4-sphere Σ, every orientation-preserving diffeomorphism f of Σ is characterized by the induced orthogonal base change automorphism of H₂(Σ; Z), which is nothing but the lift g′ of an equivalence g of the trivial surface-knot space (S⁴, F) to Σ up to composite of an identity-shift ι and smooth isotopy on Σ = S⁴(F)₂.

Proof of Theorem 2. 

For an orthogonal basis (x_∗, x′_∗) of the stable 4-sphere Σ of genus n, let (y_∗, y′_∗) be the orthogonal basis of Σ given by f_∗(x_i) = y_i and f_∗(x′_i) = y′_i for all i. By Theorem 1, there is an equivalence g of a trivial surface-knot space (S⁴, F) of genus n in S⁴ whose lift g′ to S⁴(F)₂ = Σ has g′∗(x_i) = y_i and g′∗(x′_i) = y′_i for all i. The diffeomorphisms f and g′ are homotopic since Σ is a simply connected closed oriented 4-manifold, and f and g′ induce the same intersection form on the second homology group H₂(Σ; Z), [8]. Then there is an identity-shift ι of Σ = S⁴(F)₂ such that the composite fι is smoothly isotopic to g’, [1] (Theorem 3). This completes the proof of Theorem 2. □

In the piecewise-linear category, every piecewise-linear self-homeomorphism of the 4-disk keeping the boundary identical is piecewise-linearly ∂-relatively isotopic to the identity, which is well-known as the Alexander trick, as noted in [1]. Thus, the following corollary is obtained.

Corollary 1. 

For every stable 4-sphere Σ, every orientation-preserving piecewise-linear self-homeomorphism f of Σ is characterized by the induced orthogonal base change automorphism of H₂(Σ; Z), which is nothing but the lift g′ of an equivalence g of the trivial surface-knot space (S⁴, F) to Σ up piecewise-linear isotopy of Σ.

4. Application to TOP Stable 4-Sphere

A TOP stable 4-sphere of genus n is a topological 4-manifold X = X(n) which is homeomorphic to the stable 4-sphere Σ(n). An orthogonal basis (x_∗, x′_∗) of X(n) is similarly defined by the intersection form on X(n) and sent to an orthogonal basis of Σ(n) by the homeomorphism. A TOP trivial surface-knot space of genus n is a topological pair (Y, F^Y) homeomorphic to a trivial surface-knot space (S⁴, F) of genus n. Then Y is a TOP 4-sphere and F^Y is a TOP surface-knot in Y which bounds a TOP handlebody in Y, the pullback of a handlebody bounded by F in S⁴. A TOP O2-handle pair on F^Y is the pullback of an O2-handle pair on the trivial surface-knot F in S⁴. Fix the orientation of the 4-sphere (Y, F^Y) inherited from the orientation of (S⁴, F). The following theorem is a TOP version of Theorem 1.

Theorem 3. 

For every TOP stable 4-sphere X(n), there is a TOP trivial surface-knot space (Y, F^Y) of genus n whose double branched covering space Y(F^Y)₂ = X(n). Even if X is a smooth 4-manifold, unless X(n) is not diffeomorphic to the stable 4-sphere Σ(n), the TOP trivial surface-knot space (Y, F^Y) is not smooth-able. Every orthogonal basis (x_∗, x′_∗) of X(n) is represented by a TOP O2-sphere basis (S(E_∗^Yx), S(E′_∗^Yx)) of X(n) constructed from a TOP O2-handle basis (E_∗^Yx ×I, E′_∗^Yx ×I) on F^Y in Y. For any two orthogonal bases (x_∗, x′_∗) and (y_∗, y′_∗) of X(n), there are TOP O2 -handle bases (E_∗^Yx ×I, E′_∗^Yx ×I) and (E_∗^Yy ×I, E′_∗^Yy ×I) on F^Y in Y which are transformed into each other by a TOP equivalence g of (Y, F^Y) such that (x_∗, x′_∗) and (y_∗, y′_∗) are represented by the TOP O2-sphere bases (S(E_∗^Yx), S(E′_∗^Yx)) and (S(E_∗^Yy), S(E′_∗^Yy)), respectively, which are transformed into each other by the lift g’ of g to Y(F^Y)₂ = X(n).

Proof of Theorem 3. 

Since X(n) is a TOP stable 4-sphere of genus n, there is an orientation-preserving homeomorphism h: X(n) → Σ(n). Let (S⁴, F) be a trivial surface-knot space of genus n with S⁴(F)₂ = Σ(n). For the nontrivial covering involution α of S⁴(F)₂, let α^h = h⁻¹αh be a TOP involution on X(n) with the fixed-point set h⁻¹(F). Let Y be the orbit space of X by α^h, admitting a homeomorphism h^Y: Y → S⁴ induced from h by regarding S⁴ as the orbit space of S⁴(F)₂ by α. The projection p^Y: X(n) → Y defines the double branched covering projection of Y branched along the projection image surface F^Y of h⁻¹(F), and X(n) = Y(F^Y)₂. In fact, the homeomorphism h: X(n) → Σ(n) defines a homeomorphism h^Y: (Y, F^Y) → (S⁴, F) and is considered as the lift Y(F^Y)₂ → S⁴(F)₂ of h^Y. The pair (Y, F^Y) is a TOP trivial surface-knot space of genus n sent by h^Y from the trivial surface-knot space (S⁴, F) of genus n. For any orthogonal basis (x_∗, x′_∗) of X(n), let (x_∗^S, x′_∗^S) be the orthogonal basis of Σ(n) given by x_i^S = h_∗(x_i), x′_i^S = h_∗(x′_i) (i = 1,2, ..., n). By Theorem 1, (x_∗^S, x′_∗^S) is represented by the O2-sphere basis (S(E_∗^x), S(E′_∗^x)) of Σ(n) constructed from an O2-handle basis (E_∗^x× I, E′_∗^x × I) on F in S⁴. Then (x_∗, x′_∗) is represented by the TOP O2-sphere basis (h⁻¹S(E_∗^x), h⁻¹S(E′_∗^x)) of X(n) constructed from the TOP O2 handle basis ((h^Y)⁻¹E_∗^x × I, (h^Y)⁻¹E′_∗^x × I) on F^Y in Y. For another orthogonal basis (y_∗, y′_∗) of X(n), let (y_∗^S, y′_∗^S) be the orthogonal basis of Σ(n) given by y_i^S = h_∗(y_i), y′_i^S = h_*(y′_i) (i = 1, 2, ..., n), which is represented by the O2-sphere basis (S(E_∗^y), S(E′_∗^y)) of Σ(n) constructed from an O2-handle basis (E_∗^y × I, E′_∗^y × I) on F in S⁴. Then there is an equivalence f of (S⁴, F) sending (E_∗^x × I, E′_∗^x × I) to (E_∗^y × I, E′_∗^y × I) whose lift f’ to S⁴(F)₂ = Σ(n) sends (S(E_∗^x), S(E′_∗^x)) to (S(E_∗^y), S(E′_∗^y)). Then the composite TOP equivalence g = (h^Y)⁻¹fh^Y of (Y, F^Y) sends the TOP O2 -handle basis ((h^Y)⁻¹E_∗^x × I, (h^Y)⁻¹E′_∗^x × I) on F^Y in Y to the TOP O2 -handle basis ((h^Y)⁻¹E_∗^y× I, (h^Y)⁻¹E′_∗^y × I) on F^Y in Y. The lift g’ = h⁻¹fh of g to X(n) sends the TOP O2-sphere basis (h⁻¹S(E_∗^x), h⁻¹S(E′_∗^x)) of X(n) to the TOP O2-sphere basis (h⁻¹S(E_∗^y), h⁻¹S(E′_∗^y)) of X(n) and thus sends (x_∗, x′_∗) to (y_∗, y′_∗). Suppose that X(n) is a smooth 4-manifold obtained from a smooth surface-knot space (Y, F^Y), but not diffeomorphic to Σ(n). Then there is an orientation-preserving diffeomorphism g from Y to the 4-sphere S⁴, [1]. Since the image G = gF^Y is a smooth surface-knot in S⁴ with fundamental group π₁(S⁴\G, x₀) an infinite cyclic group, the surface-knot G is a trivial surface-knot of genus n, and G is equivalent to F, [3,5]. Thus, X(n) is diffeomorphic to Σ(n), a contradiction. Thus, the TOP surface-knot space (Y, F^Y) cannot be smooth. By writing ((h^Y)⁻¹E_∗^x × I, (h^Y)⁻¹E′_∗^x × I), ((h^Y)⁻¹E_∗^y× I, (h^Y)⁻¹E′_∗^y × I), (h⁻¹S(E_∗^x), h⁻¹S(E′_∗^x)), (h⁻¹S(E_∗^y), h⁻¹S(E′_∗^y)) as (E_∗^Yx × I, E′_∗^Yx × I), (E_∗^Yy× I, E′_∗^Yy × I), (S(E_∗^Yx), S(E′_∗^Yx)), (S(E_∗^Yy), S(E′_∗^Yy)), respectively, the proof of Theorem 3 is complete. □

It is known that there are smooth stable 4-spheres X(n), not diffeomorphic to Σ(n), for infinitely many n [9]. Also, it is known that for every TOP stable 4-sphere X(n), any two homotopic self-homeomorphisms of X(n) are isotopic by TOP isotopy of X(n), [10]. Thus, every orientation-preserving self-homeomorphism of X(n) is characterized by the induced orthogonal base change automorphism of H₂(X(n); Z), which is nothing but the lift g’: X(n) → X(n) of a TOP equivalence g of the TOP trivial surface-knot (Y, F^Y) of genus n up to TOP isotopy.

5. Conclusions

For every orthogonal basis (x_∗, x′_∗) of the stable 4-sphere, Σ(n) is realized by the orthogonal basis ([S(E_∗)], [S(E′_∗)]) for an O2-handle basis (E_∗ × I, E′_∗ × I) on a trivial surface-knot space (S⁴, F) of genus n whose double branched covering space S⁴(F)₂ is Σ(n) by Lemma 1. Then it is shown that any two O2-handle bases on F can be transformed into each other by an equivalence of the surface-knot space (S⁴, F) [5]. It is concluded that any two orthogonal bases of Σ are transformable into each other by the lift f′ of f to Σ = S⁴(F)₂ (Theorem 1). Any two homotopic diffeomorphisms of Σ are smoothly isotopic if one diffeomorphism is replaced by the composition with an identity-shift [1] (Theorem 3). For this proof, Gabai’s 4D light bulb theorem is used [8]. By combining this result with Theorem 1, it is concluded that every orientation-preserving diffeomorphism of Σ is nothing but he lift g′ of an equivalence g of the trivial surface-knot space (S⁴, F) to Σ = S⁴(F)₂ up to composition of an identity-shift and smooth isotopy (Theorem 2). Here the question of whether every identity-shift is smoothly isotopic to the identity remains an unsolved problem, [1]. In the piecewise-linear category and TOP category, this problem is not needed and the arguments proceed well, except that even if a TOP stable 4-sphere X(n) is a smooth 4-manifold, unless X(n) is diffeomorphic to the stable 4-sphere Σ(n), the TOP trivial surface-knot space (Y, F^Y) of genus n with Y(F^Y)₂ = X cannot be smooth (Corollary 1 and Theorem 3).