Smooth Homotopy 4-Sphere

AKIO KAWAUCHI

Osaka Central Advanced Mathematical Institute,

Osaka Metropolitan University,

Sugimoto, Sumiyoshi-ku, Osaka 558-8585,

JAPAN

isotopic if one diffeomorphism allows a local diffeomorphism change, so that they are smoothly

concordant and piecewise-linearly isotopic.

Key-Words: - 4-sphere, Smooth homotopy 4-sphere, Trivial surface-knot, O2-handle pair, O2-sphere pair.

Received: May 5, 2023. Revised: August 7, 2023. Accepted: August 26, 2023. Published: September 26, 2023.

1 Introduction

This paper is a paper created by adding proof to the

m-manifold Xn(0) obtained from X by removing the

interiors of n mutually disjoint m-balls in the

interior of X, where choices of the m-balls are

independent of the diffeomorphism type of Xn(0).

The m-manifold X1(0) is written as X(0). By this

convention, a homotopy 4-sphere is a 4-manifold M

homotopy equivalent to the 4-sphere S4, and a

homotopy 4-ball is a 1-punctured manifold M(0) of a

homotopy 4-sphere M. Ever since the positive

solution of Topological 4D Poincaré Conjecture

affirmatively (see Corollary 1.3). The Piecewise-

Linear 4D Poincaré Conjecture is also confirmed

affirmatively by a compatible smoothability of a

compact piecewise-linear 4-manifold, [5], [6], using

Γn=Diff+(Sn-1))/rDiff+(Dn))=0 (n≦4), [7]. It is also

confirmed by an argument parallel to the Smooth

4D Poincaré Conjecture using the basic fact that

every piecewise-linear auto-homeomorphism of the

4-disk keeping the boundary identically is

piecewise-linearly ∂--relatively isotopic to the

genus n is the 4-manifold

which is the union of the n-punctured manifold

(S4)n(0) of the 4-sphere S4 and the 1-punctured

manifolds (S2×S2i)(0) (i=1,2,...,n) of the 2-sphere

products S2×S2i (i=1,2,...,n) pasting the boundary 3-

spheres of (S4)n(0) to the boundary 3-spheres of

sphere Σ of genus n is the system (S*,S’*) of disjoint

O2-sphere pairs (Si,S’i) (i=1,2,...,n) in Σ. Let

N(Si,S’i) be a regular neighborhood of the union Si

4-manifolds N(Si,S’i) (i=1,2,...,n) are mutually

disjoint and diffeomorphic to the 1-punctured

manifold (S2×S2)(0) of S2×S2. The region of a

in Σ. Since Σ is a simply connected 4-manifold, the

region Ω(S*,S’*) in Σ does not depend on choices of

the 1-handles and is uniquely determined by the

pseudo-O2-sphere basis (S*,S’*) up to isotopies of Σ,

[17]. The residual region of Ω(S*,S’*) in Σ is the 4-

manifold Ωc(S*,S’*)=cl(Σ

homotopy 4-ball shown by van Kampen theorem

and a homological argument. An O2-sphere basis of

the stable 4-sphere Σ of genus n is a pseudo-O2-

sphere basis (S*,S’*) of Σ such that the residual

(R*,R’*) and (S*,S’*) of the stable 4-sphere Σ of any

genus n

1, there is an orientation-preserving

diffeomorphism h:Σ

Σ sending (Ri,R’i) to (Si,S’i)

for all i (i=1,2,...,n).

Since the image of an O2-sphere basis (R*,R’*) of Σ

by an orientation-preserving diffeomorphism f:Σ →

is obtained from the existence of the standard O2-

sphere basis and Theorem 1.1.

Poincaré Conjecture) is a consequence of Corollary

1.2.

diffeomorphic to the 4-sphere S4.

Proof of Corollary 1.3. It is known that there is an

orientation-preserving diffeomorphism κ:M#Σ → Σ

from the connected sum M#Σ onto Σ for a positive

integer n by, [18]. The connected sum M#Σ is taken

Theorem 1.1 is equivalent to the Smooth 4D

Poincaré Conjecture. The diffeomorphism M → S4

in Corollary 1.3 made orientation-preserving is

concordantly unique since Γ5=0 by [10], [20], and

piecewise-linear-isotopically unique by, [8], [9].

However, at present it appears unknown whether it

is isotopically unique, [5], [6], [21], [22].

1.3 says further that for every closed simply

connected signature-zero spin 4-manifold M with

diffeomorphism κ:M #Σ(n) → Σ(m+n) for some n.

Then there is also a homeomorphism M→Σ(m) by

[2], [3]. It should be noted that the present technique

used for the proof of Theorem 1.1 cannot be directly

generalized to the case of m>0. In fact, it is known

paper.

N(ki)=S1×D3i (i=1,2,...,n) a system of mutually

disjoint regular neighborhoods of ki (i=1,2,...,n) in

S1×D3i with D2×S2i for every i is a homotopy 4-

sphere by van Kampen theorem and H2(M;Z)=0.

to observe that the stable 4-sphere Σ of genus n is

basis (x*,x’*)={(xj,x’j)|j=1,2,...,n} is realized by a

system of oriented simple loop pairs (e*,e’*)=

loop basis of F. For a surface-knot F in S4, an

element x in H1(F;Z) is spin if the Z2-reduction [x]2

function η:H1(F;Z2) → Z2 associated with a surface-

knot F in S4, which is defined as follows: For a

simple loop e in F bounding a surface De in S4 with

Arf invariant of the Z2-quadratic function η:H1(F;Z2)

handle D×I on F embedded in S4 such that

with 0 as the center and D×0 is called the core of the

2-handle D×I and identified with D. For a 2-handle

D×I on F in S4, the loop ∂D of the core disk D is a

spin loop in F since η([∂D]2)=0. To save notation, if

handle D×I on a surface F to a 4-manifold X, then

the 2-handle image h(D×I) and the core image h(D)

just one point q and the intersection

Q={q}×I×I, [31]. For a trivial surface-knot F of

genus n in S4, an O2-handle basis of F is a system

system (∂D*,∂D’*) given by {(∂Di,∂D’i)|i=1,2,...,n}

forms a spin loop basis of F. Every trivial surface-

knot F in S4 is moved into the boundary of a

basis as the loop basis (∂D*,∂D’*). This is because

there is an equivalence f:(S4,F)

standard spin loop basis to any given spin loop basis

of F and hence there is an O2-handle basis of F in S4

with the given spin loop basis which is the image of

the standard O2-handle basis of F by, [31], [32].

Further, any O2-handle basis of F in S4 with an

attaching part fixed is unique up to orientation-

preserving diffeomorphisms of S4 keeping F point-

wise fixed, [33].

diffeomorphism f:S4(F)2

pair system (S(D*),S(D’*)) with S(Di)=Di

S(D’i)=D’i

αD’i (i=1,2,...,n) to the standard O2-

sphere basis (S2×1*,1×S2*) of the stable 4-sphere Σ

of genus n. In particular, the 2-sphere pair system

(S(D*),S(D’*)) is an O2-sphere basis of Σ.

Proof of Lemma 2.1. Let Ai (i=1,2,...,n) be mutually

disjoint 4-balls which are regular neighborhoods of

the 3-balls Di×I∪D’i×I (i=1,2,...,n) in S4. The

space S4(S2)2 is diffeomorphic to S4, the double

branched covering space (S4)n(0)(P)2 of (S4)n(0)

branched along P is diffeomorphic to (S4)n(0). On the

other hand, for the proper surface Pi=F

ball Ai, the pair (Ai,Pi) is considered as a1-punctured

pair of a trivial torus-knot space (S4,T), so that the

double branched covering space Ai(Pi)2 is

diffeomorphic to the 1-punctured manifold of the

double branched covering space S4(T)2. The trivial

torus-knot space (S4,T) is the double of the product

the boundary ∂(B(o)2×I2), where B(o)2 is the double

branched covering space of B branched along o

which is diffeomorphic to the product S2×I. This

means that the 5-manifold B(o)2×I2 is the product

S2×I3 and S4(T)2 is diffeomorphic to S2×S2, which

shows that Ai(Pi)2 is diffeomorphic to (S2×S2)(0). This

construction also shows that there is an orientation-

standard O2-sphere pair (S2×1i,1×S2i) of the

desired orientation-preserving diffeomorphism f:

S4(F)2 → Σ is constructed from a diffeomorphism

The identification of S4(F)2=Σ is fixed by an

D’*×I) and (E*×I,E’*×I) of a trivial surface-knot F

of genus n in S 4, there is an orientation-preserving

α-equivariant diffeomorphism f” of Σ sending the 2-

sphere pair system (S(D*),S(D’*)) to the 2-sphere

pair system (S(E*),S(E’*)). In particular, the 2-

O2-handle basis (D*×I, D’*×I) of F to the O2-handle

basis (E*×I,E’*×I) of F by uniqueness of an O2-

handle pair, [31], [33]. By construction, the lifting

diffeomorphism f”:S4(F)2 → S4(F)2 of f sends

the 2-sphere pair system (S(D*),S(D’*)) for every

O2-handle basis (D*×I,D’*×I) is shown to be an O2-

sphere basis of Σ by taking (E*×I, E’*×I) a standard

O2-handle basis of F. This completes the proof of

Corollary 2.2.

denote the centerline system of the band system b*.

The following lemma shows that there is a canonical

n-rooted disk family (d,d*,b*) associated with an O2-

handle basis (D*×I,D’*×I) of a trivial surface-knot F

of genus n in S4.

basis of a trivial surface-knot F of genus n in S4, and

(d,d*,b*) an n-rooted disk family. Then there is an

embedding

(3) There is an identification

as 2-handle systems on F.

Lemma 2.3 says that the 2-handle systems D*×I and

D’*×I are attached to a handlebody V bounded by F

O2-handle basis of F, then it is easy to construct an

embedding φ’: (d,d*,b*)×I → (S4,E*×I,E’*×I) with

(1)-(3) taking φ’ and (E*×I,E’*×I) as φ and

equivalence f: (S4,F) → (S4,F) keeping F set-wise

Lemma 2.3.

In Lemma 2.3, the embedding φ, the 3-ball B=

respectively called a bump embedding, a bump 3-

ball, a bump handlebody and a bump pair of F in S4.

For a bump embedding

there is an embedding φ”:d×I → S4(F)2 with pφ”=φ.

Since p(φ”(d*×I),φ”(b*×I))=(D*×I,D’*×I) by the

conditions (1)-(3) of Lemma 2.3, the images

φ”(d*×I) and φ”(b*×I) are respectively considered

φ”:(d,d*,b*)×I → (S4(F)2,D*×I,D’*×I)

with pφ”=φ. This embedding is called a lifting

bump embedding of the bump embedding φ. In this

handlebody φ”(cl(d

is identical to the lifting bump embedding

φ”:(d,d*,b*)×I→(Σ,D*×I,D’*×I).

Proof of Lemma 2.4. The 0-section (d,d*,b*)×0 of

family (d,d*,b*) is identified with (d,d*,b*). Move the

disk uφ”(d) into the disk φ”(d) in Σ and then move

the disk system uφ”(d*) and the band system

uφ”(b*) into the disk system φ”(d*) and the band

system φ”(b*) in the disk φ”(d), respectively. These

not meet the bump 3-ball B=φ”(d×I) in Σ. In fact,

3 Proof of Theorem 1.1

all i. Let (B,V) be a bump pair for the O2-handle

basis (D*×I,D’*×I) of F in S4 defined soon after

Lemma 2.3. Recall that the two lifts of (B,V) to Σ

under the double branched covering projection

p:S4(F)2 → S4 are given by (B,V) and (αB,αV). To

embedding gu:Σ(0) → Σ preserves the bump pair

(B,V) in Σ identically and has the property that

every disk interior of the disk systems guαD* and

guαD’* meets every disk interior of the disk systems

number 0.

Proof of Lemma 3.2. By Lemma 2.4, there is a

diffeomorphism g:Σ → Σ which is isotopic to the

identity such that the composite embedding gu:Σ(0)

systems αD* and αD’* is a disjoint pair, every disk

interior of the disk systems guαD* and guαD’* meets

every disk interior of the disk systems αD* and αD’*

only with intersection number 0 by the homological

identities [guS(Di)]=[S(Di)], [guS(D’i)] =[S(D’i)] for

This completes the proof of Lemma 3.2.