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FAMILIES OF DIFFEOMORPHISMS AND CONCORDANCES DETECTED BY TRIVALENT GRAPHS BORIS BOTVINNIK AND TADAYUKI WATANABE Abstract. We study families of diffeomorphisms detected by trivalent graphs via the Kontsevich classes. We specify some recent results and constructions of the second named author to show that those non-trivial elements in homotopy groups π*(BDiff (D d )) Q are lifted to homotopy groups of the moduli space of h -cobordisms π*(BDiff t(D d × I )) Q . As a geometrical application, we show that those elements in π*(BDiff (D d )) Q for d 4 are also lifted to the rational homotopy groups π*(M psc (D d ) h 0 ) Q of the moduli space of positive scalar curvature metrics. Moreover, we show that the same elements come from the homotopy groups π*(M psc t (D d × I ; g0) h 0 ) Q of moduli space of concordances of positive scalar curvature metrics on D d with fixed round metric h0 on the boundary S d-1 . Contents 1. Results 2 1.1. Extension of graph surgery to concordance 2 1.2. Application to the moduli space of psc-metrics 4 1.3. Conventions 6 2. Graph surgery 6 3. Alternative definition of Y-surgery by framed links 8 3.1. Framed link for Type I surgery 8 3.2. Family of framed links for Type II surgery 9 3.3. Hopf link surgery for links 10 3.4. Type I Y-surgery for links 13 3.5. Type II Y-surgery for links 16 4. Family of framed links for graph surgery 18 5. Bordism modification to a S k(d-3) -family of surgeries 24 5.1. From a B Γ -family to a S k(d-3) -family 24 5.2. Modification into a family of h -cobordisms 26 6. Odd dimensional case 27 7. Proof of Theorems 1.7 and 1.9 28 7.1. Recollection: admissible Morse functions 28 7.2. Recollection: surgery for families of Morse functions 29 7.3. Back to the proof 31 References 31 Date : January 28, 2022. 2000 Mathematics Subject Classification. 57M27, 57R57, 58D29, 58E05, 53C27, 57R65, 58J05, 58J50. BB was partially supported by Simons collaboration grant 708183. TW was partially supported by JSPS Grant-in-Aid for Scientific Research 21K03225 and by RIMS, Kyoto University. 1 arXiv:2201.11373v1 [math.GT] 27 Jan 2022

Transcript of FAMILIES OF DIFFEOMORPHISMS AND CONCORDANCES …

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FAMILIES OF DIFFEOMORPHISMS AND CONCORDANCES

DETECTED BY TRIVALENT GRAPHS

BORIS BOTVINNIK AND TADAYUKI WATANABE

Abstract. We study families of diffeomorphisms detected by trivalent graphs via the Kontsevichclasses. We specify some recent results and constructions of the second named author to show thatthose non-trivial elements in homotopy groups π∗(BDiff∂(Dd)) ⊗Q are lifted to homotopy groupsof the moduli space of h -cobordisms π∗(BDifft(Dd × I)) ⊗ Q . As a geometrical application, weshow that those elements in π∗(BDiff∂(Dd))⊗Q for d ≥ 4 are also lifted to the rational homotopygroups π∗(M

psc∂ (Dd)h0) ⊗ Q of the moduli space of positive scalar curvature metrics. Moreover,

we show that the same elements come from the homotopy groups π∗(Mpsct (Dd × I; g0)h0) ⊗ Q of

moduli space of concordances of positive scalar curvature metrics on Dd with fixed round metrich0 on the boundary Sd−1 .

Contents

1. Results 2

1.1. Extension of graph surgery to concordance 2

1.2. Application to the moduli space of psc-metrics 4

1.3. Conventions 6

2. Graph surgery 6

3. Alternative definition of Y-surgery by framed links 8

3.1. Framed link for Type I surgery 8

3.2. Family of framed links for Type II surgery 9

3.3. Hopf link surgery for links 10

3.4. Type I Y-surgery for links 13

3.5. Type II Y-surgery for links 16

4. Family of framed links for graph surgery 18

5. Bordism modification to a Sk(d−3) -family of surgeries 24

5.1. From a BΓ -family to a Sk(d−3) -family 24

5.2. Modification into a family of h-cobordisms 26

6. Odd dimensional case 27

7. Proof of Theorems 1.7 and 1.9 28

7.1. Recollection: admissible Morse functions 28

7.2. Recollection: surgery for families of Morse functions 29

7.3. Back to the proof 31

References 31

Date: January 28, 2022.2000 Mathematics Subject Classification. 57M27, 57R57, 58D29, 58E05, 53C27, 57R65, 58J05, 58J50.BB was partially supported by Simons collaboration grant 708183.TW was partially supported by JSPS Grant-in-Aid for Scientific Research 21K03225 and by RIMS, Kyoto

University.

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2 BORIS BOTVINNIK AND TADAYUKI WATANABE

1. Results

1.1. Extension of graph surgery to concordance. Let Diff∂(Dd) be the group of diffeomor-

phisms φ : Dd → Dd which restrict to the identity near the boundary ∂Dd = Sd−1 .

Recently, the second author obtained the following theorem.

Theorem 1.1 ([Wa09, Wa18a, Wa18b, Wa21]). Let d ≥ 4. For each k ≥ 2, the evaluation of

Kontsevich’s characteristic classes on Dd -bundles gives an epimorphism

πk(d−3)BDiff∂(Dd)⊗Q→ Aeven/oddk

to the space of Aeven/oddk of trivalent graphs. For k = 1, the same result holds for the group

π2n−2BDiff∂(D2n+1)⊗Q for many odd integers d = 2n+1 ≥ 5 satisfying some technical condition1.

Theorem 1.1 was proved by evaluating Kontsevich’s characteristic classes ([Kon]) on elements

constructed by surgery on trivalent graphs embedded in Dd .

Here we recall the definition of the spaces Aeven/oddk of connected trivalent graphs, which are

the trivalent parts of Kontsevich’s graph homology [Kon]. In general, trivalent graph has even

number of vertices, and if it is 2k , then the number of edges is 3k . Let V (Γ) and E(Γ) denote

the sets of vertices and edges of a trivalent graph Γ, respectively. Labellings of a trivalent graph Γ

are given by bijections V (Γ) → 1, 2, . . . , 2k , E(Γ) → 1, 2, . . . , 3k . Let Gk be the vector space

over Q spanned by the set G 0k of all labelled connected trivalent graphs with 2k vertices modulo

isomorphisms of labelled graphs. The version A evenk , which works for even-dimensional manifolds,

is defined by

A evenk = Gk/IHX, label change,

where the IHX relation is given in Figure 1

Figure 1. IHX relation.

and the label change relation is generated by the following relations:

Γ′ ∼ −Γ, Γ′′ ∼ Γ.

Here, Γ′ is the graph obtained from Γ by exchanging labels of two edges, Γ′′ is the graph obtained

from Γ by exchanging labels of two vertices. The version A oddk , which works for odd-dimensional

manifolds, can be defined similarly as A evenk except a small modification in the orientation con-

vention. Namely, let Gk be the vector space over Q spanned by the set G 0k of all pairs (Γ, o) of

1d = 5, 7, 9, 11, 15, 19, 23, 24, 25, . . . , checked by non-integrality of some rational numbers involving the Bernoulli

numbers in [Wa09]. Actually, this holds for all d ≥ 5 odd ([KrRW]). See also Remark 1.6

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labelled connected trivalent graphs Γ with 2k vertices modulo isomorphisms of labelled graphs and

orientations o of the real vector space H1(Γ;R). Then we define

A oddk = Gk/IHX, label change, orientation reversal

where the IHX and the label change relation is the same as above, and the orientation reversal is

the following:

(Γ,−o) ∼ −(Γ, o).

Let X be a d-dimensional path connected smooth manifold with non-empty boundary. Let

Difft(X × I) := Diff(X × I,X × 0 ∪ ∂X × I) be the group of pseudoisotopies. There is a

natural fiber sequence

(1) Diff∂(X × I)i−→ Difft(X × I)

∂−→ Diff∂(X × 1),

where i : Diff∂(X × I) → Difft(X × I) is the inclusion, and ∂ : Difft(X × I) → Diff∂(X × 1)restricts a diffeomorphism ψ : X× I → X× I to the top part of the boundary ψ|X×1 . This gives

a corresponding fiber sequence of the classifying spaces

(2) BDiff∂(X × I)i−→ BDifft(X × I)

∂−→ BDiff∂(X × 1).

Remark 1.2. The group of pseudoisotopies Difft(X × I) is often denoted as C∂(X).

The first main result of this paper is the following.

Theorem 1.3 (Theorem 2.3, 6.1). Let d ≥ 4. All the elements given by surgery on trivalent graphs

are in the image of the homomorphism

∂∗ : πk(d−3)BDifft(X × I)→ πk(d−3)BDiff∂(X).

Furthermore, if d is even (resp. if d is odd and d = 2m + 1), then each element in the group

πk(d−3)BDiff∂(X) ⊗ Q constructed by surgery on a trivalent graph embedded in X has a lift in

πk(d−3)BDifft(X × I) ⊗ Q represented by a smooth (X × I)-bundle E → Sk(d−3) that admits

fiberwise Morse functions with only critical loci of indices 1 and 2 (resp. indices m and m+ 1).

Remark 1.4. A version of this theorem for bordism group was pointed out to the second author by

Peter Teichner ([Wa20, Theorem 9.3]). We would like to emphasize the following new features in

Theorem 1.3:

(1) All the bordism classes in ΩSOk(d−3)(BDifft(X×I)) and ΩSO

k(d−3)(BDiff∂(X)) given by surgery

on trivalent graphs in X × 1 and X , respectively, are represented by families of handle-

bodies parametrized by Sk(d−3) with only 1- and 2-handles. This point is crucial in our

applications to rational homotopy groups of the moduli spaces of metrics of positive scalar

curvature and requires some further work.

(2) We describe in this paper the details about the interpretation of the graph surgery in terms

of spherical modifications along framed Hopf links, which were sketched in [Wa20, §9]. This

could also be applied to constructions of families of embeddings in a manifold.

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Corollary 1.5. Let d ≥ 4 and k ≥ 2. If d is even (resp. if d is odd), then the group

πk(d−3)BDifft(Dd × I) ⊗ Q is nontrivial whenever A evenk (resp. A odd

k ) is nontrivial. For k = 1,

π2n−2BDifft(D2n+1 × I) ⊗ Q is nontrivial for many odd integers d = 2n + 1 ≥ 5 satisfying the

same technical condition as in Theorem 1.1.

Remark 1.6. (1) Note that this includes results for pseudoisotopies of D4 . It was proved in

[Wa18b] that π2(BDifft(D4 × I)) ⊗ Q is nonzero. Theorem 1.3 shows that the groups

πkBDifft(D4 × I)⊗Q are non-trivial for many k > 2. This is new result.

(2) Recently, A. Kupers and O. Randal-Williams ([KuRW]), M. Krannich and O. Randal-

Williams ([KrRW]) computed the rational homotopy groups of BDiff∂(Dd) in some wide

range of dimensions surprisingly completely. In particular, it follows from their results that

for n > 5, the natural map

π2n−2BDifft(D2n+1 × I)⊗Q→ π2n−2BDiff∂(D2n+1)⊗Q

is an isomorphism and both terms are isomorphic to Q ⊕ (K2n−1(Z) ⊗ Q). In particular,

Corollary 1.5 for d = 2n+ 1 > 11 and k = 1 follows from their results.

1.2. Application to the moduli space of psc-metrics. Let h0 be the standard round metric

on Sd−1 = ∂Dd , and R∂(Dd)h0 be the space of Riemannian metrics g on the disk Dd which have

a form h0 + dt2 near the boundary Sd−1 . The group Diff∂(Dd) acts on R∂(Dd)h0 by pulling a

metric back: g · φ 7→ φ∗g . It is easy to see that this action is free, and, since the space R∂(Dd)h0

is contractible, there is a homotopy equivalence

BDiff∂(Dd) ∼M∂(Dd)h0 := R∂(Dd)h0/Diff∂(Dd).

Thus the moduli space M∂(Dd)h0 could be thought as a geometrical model of the classifying

space BDiff∂(Dd). Below we identify the spaces M∂(Dd)h0 and BDiff∂(Dd). Let Rpsc(Dd)h0 ⊂R∂(Dd)h0 be a subspace of metrics with positive scalar curvature (which will abbreviated as “psc-

metrics”). We have the following diagram of principal Diff∂(Dd)-fiber bundles:

Rpsc∂ (Dd)h0 R∂(Dd)h0

M psc∂ (Dd)h0 M∂(Dd)h0

-i

?

p

?

p

Here M psc∂ (Dd)h0 := Rpsc

∂ (Dd)h0/Diff∂(Dd) is the moduli space of psc-metrics.

Theorem 1.7. Let d ≥ 4 be an integer. All classes given by surgery on trivalent graphs are in the

image of the induced map

ι∗ : πqMpsc∂ (Dd)h0 ⊗Q→ πqBDiff∂(Dd)⊗Q.

Hence, all nontrivial elements of πqBDiff∂(Dd) ⊗ Q given by surgery on trivalent graphs lift to

nontrivial elements of πqMpsc∂ (Dd)h0 ⊗Q.

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Remark 1.8. For d ≥ 6, Theorem 1.7 follows also from [E-RW, Theorem F]. We give a geometrical

proof of Theorem 1.9 which, in particular, proves Theorem 1.7. In fact, it is not difficult to prove

a stronger statement for the existence of the lift in the moduli space. Namely, all classes given by

surgery on trivalent graphs are in the image of the induced map ι∗ : πqMpsc∂ (X)h0 → πqBDiff∂(X)

for an arbitrary smooth manifold X of dimension d ≥ 4 having a psc metric h0 .

Next, we fix some geometrical data. Consider the subset (Dd×0)∪ (Sd−1× I) ⊂ Dd× I and

fix a psc-metric g0 ∈ Rpsc(Dd × 0)h0 . We view the cylinder Dd × I as a manifold with corners.

Let U be a colar of (Dd × 0) ∪ (Sd−1 × I); we assume that U is parametrized by (x, t, s) near

the corner Sd−1 × 0 , as it it shown in Figure 2, where x ∈ Sd−1 × 0 .Dd × 1

tsx

Figure 2. A collar of (Dd × 0) ∪ (Sd−1 × I)→ Dd × I .

We consider a subspace Rt(Dd × I; g0) ⊂ R(Dd × I) of Riemannian metrics g which restrict to

(3)

g0 + ds2 near Dd × 0h0 + ds2 + dt2 near Sd−1 × Ig + ds2 near Dd × 1 for some g ∈ R(Dd × 1)h0

Let Rpsct (Dd × I; g0)h0 ⊂ Rt(Dd × I; g0)h0 be a corresponding subspace of psc-metrics. Again,

we notice that the group Difft(Dd × I) acts freely on a contractible space Rt(Dd × I; g0)h0 . In

particular, we have homotopy equivalence

BDifft(Dd × I) ∼Mt(Dd × I; g0)h0 := Rt(Dd × I; g0)h0/Difft(Dd × I) .

Again we have the following diagram of principal Difft(Dd × I)-fiber bundles:

Rpsct (Dd × I; g0)h0 Rt(Dd × I; g0)h0

M psct (Dd × I; g0)h0 Mt(Dd × I; g0)h0

-i

?

p

?

p

We also notice that the restriction map

Rpsct (Dd × I; g0)h0 → Rpsc

∂ (Dd)h0 , g 7→ g = g|Dd×1

where g is given in (3), induces a map of corresponding moduli spaces:

∂psc : M psct (Dd × I; g0)h0 →M psc

∂ (Dd)h0

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Theorem 1.9. Let d ≥ 4 be an integer. All lifts in πqMpsc∂ (Dd)h0 ⊗Q found in Theorem 1.7 are

in the image of the homomorphism

∂psc∗ : πk(d−3)Mpsct (Dd × I; g0)h0 ⊗Q→ πk(d−3)M

psc∂ (Dd)h0 ⊗Q .

Hence, any nontrivial elements of πqBDiff∂(Dd) ⊗ Q given by surgery on trivalent graphs lift to

nontrivial elements of πk(d−3)Mpsct (Dd × I; g0)h0 ⊗Q.

1.3. Conventions.

• A framed embedding (or a framed link) consists of an embedding ϕ : S → X between

smooth manifolds and a choice of a normal framing τ on ϕ(S), where by a normal framing

we mean a trivialization ν(ϕ(S)) ∼= ϕ(S)× Rcodimϕ(S) of the normal bundle.

• We will often say “a framed embedding ϕ” or “a framed link ϕ”, instead of (ϕ, τ).

• We consider links as submanifolds equipped with parametrizations. Thus in this paper links

are embeddings. Also, we assume that famlies of links are smoothly parametrized.

• We will consider trivialities of families or bundles in several different meanings. Instead of

saying just “trivial bundle”, we will say that a bundle/family is trivialized if it is equipped

with a trivialization. If it admits at least one trivialization, we say it is trivializable. A

given family ϕs of some objects ϕs is strictly trivial if ϕs does not depend on s , i.e.,

ϕs = ϕs0 for some s0 . It seems usual to say a bundle is trivial if it is trivializable.

2. Graph surgery

We take an embedding Γ → IntX of a labeled, edge-oriented trivalent graph Γ. We put a

Hopf link of the spheres Sd−2 and S1 at the middle of each edge, as in Figure 3.

Figure 3. Decomposition of embedded trivalent graph into Y-shaped pieces.

Then every vertex of Γ gives a Y-shaped component Y-graph of Type I or and II, see Figure 4

below, i.e. an Y-graph is a vertex together with framed spheres Sd−2 and S1 attached. We call the

attached spheres leaves of a Y-graph. This construction transforms the graph Γ into 2k compo-

nents Y-graphs. We take small closed tubular neighborhoods of those Y-graphs, namely the disjoint

union of the ε-tubular neighborhoods of the leaves and the trivalent vertex (a point) connected by

ε/2-tubular neighborhoods of the edges for some small ε , and denote them by V (1), V (2), . . . , V (2k) .

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FAMILIES OF DIFFEOMORPHISMS AND CONCORDANCES 7

Figure 4. Y-graphs of Type I and II

They form a disjoint union of handlebodies embedded in IntX . A Type I Y-graph gives a handle-

body (of a Type I) which is diffeomorphic to the handlebody obtained from a d-ball by attaching

two 1-handles and one (d− 2)-handle in a standard way, namely, along unknotted unlinked stan-

dard attaching spheres in the boundary of Dd . A Type II Y-graph gives a handlebody (of a Type

II) which is diffeomorphic to the handlebody obtained from a d-ball by attaching one 1-handle and

two (d− 2)-handles in a standard way.

Let V = V (i) be one of the Type I handlebodies and let αI : S0 → Diff(∂V ), S0 = −1, 1 ,be the map defined by αI(−1) = 1 , and by setting αI(1) as the “Borromean twist” corresponding

to the Borromean string link Dd−2 ∪Dd−2 ∪D1 → Dd . The detailed definition of αI can be found

in [Wa18b, §4.5].

Let V = V (i) be one of the Type II handlebodies and let αII : Sd−3 → Diff(∂V ) be the

map defined by comparing the trivializations of of the family of complements of an Sd−3 -family

of embeddings Dd−2 ∪ D1 ∪ D1 → Dd obtained by parametrizing the second component in the

Borromean string link Dd−2∪Dd−2∪D1 → Dd with that of the trivial family of ∂V . The detailed

definition of αII can be found in [Wa18b, §4.6].

Rd−2

Dd−2

R1

D1

R1

Dd−2

Figure 5. Borromean string link Dd−2 ∪Dd−2 ∪D1 → Dd

The Borromean string link has the following important property, which will be frequently used

later.

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8 BORIS BOTVINNIK AND TADAYUKI WATANABE

Property 2.1. If one of the three components in the Borromean string link Dd−2∪Dd−2∪D1 → Dd

is deleted, then the string link given by the remaining two components is isotopic relative to the

boundary to the standard inclusion of disks.

For each i-th vertex of Γ we let Ki = S0 or Sd−3 depending on whether this vertex is of Type

I or II. Accordingly, let αi : Ki → Diff(∂V (i)) be αI or αII . Let BΓ = K1 × · · · ×K2k . By using

the families of twists above, we define

EΓ = ((X − Int (V (1) ∪ · · · ∪ V (2k)))×BΓ) ∪∂ ((V (1) ∪ · · · ∪ V (2k))×BΓ),

where the gluing map is given by

ψ : (∂V (1) ∪ · · · ∪ ∂V (2k))×BΓ → (∂V (1) ∪ · · · ∪ ∂V (2k))×BΓ

ψ(x, t1, . . . , t2k) = (αi(ti)(x), t1, . . . , t2k) (for x ∈ ∂V (i)).

Proposition 2.2 ([Wa18b]). Let X be a d-dimensional compact manifold having a framing τ0 .

The natural projection πΓ : EΓ → BΓ is an (X, ∂)-bundle, and it admits a vertical framing that is

compatible with the surgery and that agrees with τ0 near the boundary, and it gives an element of

ΩSO(d−3)k(BDiff(X, ∂)),

where BDiff(X, ∂) is the classifying space for framed (X, ∂)-bundles. We denote this element by

Ψk(Γ).

Theorem 2.3. (1) The (X, ∂)-bundle πΓ : EΓ → BΓ for an embedding φ : Γ → IntX is

related by an (X, ∂)-bundle bordism to an (X, ∂)-bundle $Γ : EΓ → Sk(d−3) obtained from

the product bundle X × Sk(d−3) → Sk(d−3) by fiberwise surgeries along a Sk(d−3) -family of

framed links hs : S1 ∪ Sd−2 → IntX , s ∈ Sk(d−3) , that satisfies the following conditions:

(a) hs is isotopic to the Hopf link for each s.

(b) The restriction of hs to Sd−2 is a constant Sk(d−3) -family.

(c) There is a small neighborhood N of Imφ such that the image of hs is included in N

for all s ∈ Sk(d−3) .

(2) There exists an (X×I)-bundle ΠΓ : WΓh → Sk(d−3) with structure group Difft(X×I) such

that

(a) the fiberwise restriction of ΠΓ to X × 1 is $Γ ,

(b) WΓh is obtained by attaching Sk(d−3) -families of (d+ 1)-dimensional 1- and 2-handles

to the product (X × I)-bundle (X × I)× Sk(d−3) → Sk(d−3) at (X × 1)× Sk(d−3) .

This is devided into Proposition 4.1, Corollary 5.4, and Proposition 5.5.

3. Alternative definition of Y-surgery by framed links

3.1. Framed link for Type I surgery. Let d ≥ 4. Let K1,K2,K3 be the unknotted spheres in

IntX that are parallel to the cores of the handles of Type I handlebody V of indices 1,1,d − 2,

respectively. Let ci be a small unknotted sphere in IntV that links with Ki with the linking

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FAMILIES OF DIFFEOMORPHISMS AND CONCORDANCES 9

number 1. Let L′1∪L′2∪L′3 be a Borromean rings of dimensions d−2, d−2, 1 embedded in a small

ball in IntV that is disjoint from

K1 ∪K2 ∪K3 ∪ c1 ∪ c2 ∪ c3.

For each i = 1, 2, 3, let Li be a knotted sphere in IntV obtained by connect summing ci and L′i

along an embedded arc that is disjoint from the cocores of the 1-handles and from other components,

so that Li ’s are mutually disjoint. Then Ki ∪ Li is a Hopf link in d-dimension. If Ki is null in

X for i = 1, 2, 3, namely, Ki bounds an embedded disk in X , then each component of the six

component link⋃3i=1(Ki ∪ Li) is an unknot in X , and we may consider it as a framed link by

canonical framings induced from the standard sphere by the isotopies along the spanning disks

(Figure 6, V (1) ). The following is a framed link definition of Type I surgery.

Definition 3.1 (Y-surgery of Type I). We define the Type I surgery on V to be the surgery along

the six component framed link⋃3i=1(Ki ∪ Li) in V .

We will see in section 3.4 (Remark 3.9) that this definition is equivalent to that we have given

in section 2. The proof is an analogue of [Ha, §2] or [GGP, Lemma 2.1].

3.2. Family of framed links for Type II surgery. Similarly, let K1,K2,K3 be the unknotted

spheres in IntX that are parallel to the cores of the handles of Type II handlebody V of indices

1,d− 2,d− 2, respectively. Let ci be a small framed unknotted sphere in IntV that links with Ki

with the linking number 1. Let L′1,s∪L′2,s∪L′3,s (s ∈ Sd−3 ) be a (d−3)-parameter family of three

component framed links of dimensions d − 2, 1, 1 with only (isotopically) unknotted components

embedded in a small ball in IntV disjoint from K1 ∪K2 ∪K3 ∪ c1 ∪ c2 ∪ c3 such that L′1,s, L′3,s

are unknotted components in V that do not depend on s , and the union of the locus of L′2,s and

L′1,s ∪ L′3,s forms a closure of the Borromean string link of dimensions d− 2, d− 2, 1.

For each i = 1, 2, 3, let Li,s be a knotted sphere in IntV obtained by connect summing ci

and L′i,s along an embedded arc that is disjoint from the cocores of the 1-handles and from other

components, so that Li,s ’s are mutually disjoint. Then Ki ∪ Li,s is a Hopf link in d-dimension. If

Ki is null in X for i = 1, 2, 3, then each component of the six component link⋃3i=1(Ki ∪ Li,s) is

fiberwise isotopic to a constant family of an unknot in X , and we may consider it as a family of

framed links by canonical framings (Figure 6, V (2) ). The following is a framed link definition of

Type II surgery.

Definition 3.2 (Y-surgery of Type II). We define the Type II surgery on V to be the Sd−3 -family

of surgeries along the family of the six component framed link⋃3i=1(Ki ∪ Li,s), s ∈ Sd−3 , in V ,

which produces a (V, ∂V )-bundle over Sd−3 .

We will see in section 3.5 (Remark 3.14) that this definition is equivalent to that we have given

in section 2.

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10 BORIS BOTVINNIK AND TADAYUKI WATANABE

Figure 6. Framed link for Θ-graph surgery

3.3. Hopf link surgery for links. We would like to describe the effect of a Y-surgery of Type

I or II when a link in the complement of the Y-graph is present. Since a Y-surgery consists of

surgeries of three Hopf links, we shall first consider the effect of a single Hopf link surgery.

3.3.1. Surgery of X on a framed link L. We shall recall the definition of surgery on a framed link

L in IntX . Consider a (d + 1)-dimensional cobordism WL obtained from X × I by attaching

disjoint handles along L×1 in X ×1 . Namely, for each framed embedding ` : Si → L×1 ,we attach (d+ 1)-dimensional (i+ 1)-handle along a small tubular neighborhood of ` . The handle

attachments can be done disjointly and simultaneously, and gives a (d+ 1)-dimensional cobordism

WL between X×0 and some d-manifold XL . We say that WL is obtained from X×I by surgery

along L , or by attaching handles along L . Let ∂tWL = X ×0∪ ∂X × I , ∂−WL = X ×0 , and

∂+WL = ∂WL \ Int ∂tWL = XL .

Figure 7. Manifold WL

3.3.2. Concordance of a cobordism. When a link c in X is present, surgery on a framed link L

in X \ c changes the pair (X, c). It may happen that surgeries for two choices (c, L) and (c′, L′)

of the links in X should be considered equivalent. Here, we consider the notion of concordance

between two such data, defined as follows.

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FAMILIES OF DIFFEOMORPHISMS AND CONCORDANCES 11

Definition 3.3. Let W be a relative cobordism between ∂−W = X × 0 and ∂+W such that

∂W = ∂+W ∪∂X×1 (∂X × I) ∪∂X×0 ∂−W.

Let ∂tW = (∂X × I) ∪∂X×0 ∂−W .

(1) A concordance of a cobordism W fixing ∂tW is a (W,∂tW )-bundle over [0, 1] whose fiber

over 0 is identified with W . We say that pairs (W,∂tW ) and (W ′, ∂tW′) of relative

cobordisms are concordant if there is a concordance q : W → [0, 1] of W fixing ∂tW such

that q−1(1) = W ′ and (∂tW )× 1 = ∂tW′ (canonical identification).

(2) For framed links L and L′ in ∂+W and ∂+W′ , respectively, a concordance between the

triples (W,∂tW,L) and (W ′, ∂tW′, L′) is a concordance q : W → [0, 1] between (W,∂tW )

and (W ′, ∂tW′) fixing ∂tW that has a trivialized subbundle L of

∂+W =⋃

t∈[0,1]

∂+q−1(t)

with a fiberwise framing such that it restricts to the framed links L and L′ on q−1(0)and q−1(1).

(3) For the triple (W,∂tW,L) as above, we define (W,∂tW )L to be the cobordism obtained

by attaching handles along the framed link L .

Figure 8. A concordance between (W,∂tW,L) and (W ′, ∂tW′, L′)

Remark 3.4. The definition of concordance for manifold pair is not as usual. Usually, the projection

proj q|L

: L→ I for the concordance L may not be level-preserving, whereas we assume so.

The following lemma is evident from definition.

Lemma 3.5. Let W,W ′, L, L′ be as in Definition 3.3.

(1) A concordance between (W,∂tW ) and (W ′, ∂tW′) fixing ∂tW induces a relative diffeo-

morphism between them .

(2) A concordance between (W,∂tW,L) and (W ′, ∂tW′, L′) fixing ∂tW induces a relative dif-

feomorphism between (W,∂tW ) and (W ′, ∂tW′) that maps L to L′ .

(3) A concordance between (W,∂tW,L) and (W ′, ∂tW′, L′) fixing ∂tW induces a concordance

between (W,∂tW )L and (W ′, ∂tW′)L′

fixing ∂tW .

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12 BORIS BOTVINNIK AND TADAYUKI WATANABE

3.3.3. Hopf link surgery. Suppose a d-manifold X is equipped with some embedded objects inside,

such as links or Y-links. By a small Hopf link in X , we mean a Hopf link in a d-ball b in X with

sufficiently small radius so that b is disjoint from the given embedded objects in X .

Let K,L be the components of a Hopf link in IntX of dimensions 1, d−2 with standard framing

and with spanning disks d1, d2 in IntX , respectively. Let c1, c2 be framed spheres of dimensions

d − 2, 1, respectively, in IntX such that d1 (resp. d2 ) intersects c1 (resp. c2 ) transversally by

one point and does not intersect other component in c1 ∪ c2 nor K ∪ L (See Figure 9, left). Let

Nd1∪d2 be a small closed neighborhood of d1 ∪ d2 . Let c′1 ∪ c′2 be a framed link in IntX obtained

from c1 ∪ c2 by component-wise connect-summing a small Hopf link in Nd1∪d2 . Let K ′ ∪ L′ be

another framed Hopf link in Nd1∪d2 that is small and disjoint from c′1 ∪ c′2 . (See Figure 9, right.)

The following lemma is an analogue of [Ha, Proposition 2.2].

Lemma 3.6 (Hopf link surgery). There is a concordance q : W → [0, 1] between the triples

(WK∪L, ∂tWK∪L, c1 ∪ c2) and (WK′∪L′ , ∂tWK′∪L′ , c′1 ∪ c′2) such that the restriction of q to ((X \

IntNd1∪d2)× 1)× [0, 1] is trivial.

Figure 9. Surgery along Hopf link K ∪ L

Proof. Let L1 = L and L2 = K . Before going to the proof, we define band-sums ci#Li . We

choose an embedded path γi in di that goes from di ∩ ci to di ∩ Li . Then we may connect-sum

ci with Li along γi so that the result is disjoint from di . More precisely, the restriction of the

normal bundle of di on γi is an Rdim ci -bundle. Thus γi can be thickened to a dim ci -disk bundle

in Nd1∪d2 that is perpendicular to di and its restriction on the endpoints are dim ci -disks in ci

and Li . The disk bundle is a (dim ci + 1)-dimensional 1-handle attached to ci ∪ Li along which

surgery can be performed. This surgery produces the connected sum ci#Li along γi whose result

is disjoint from di . See Figure 10, left.

Now let us return to the framed link K ∪ L ∪ c1 ∪ c2 . We perform surgeries on the link

K ∪L = L2 ∪L1 , then the component ci can be slid over γi and the (dimLi + 1)-handle attached

to Li . The result of the handle slide is ci#Li defined as in the previous paragraph, where Li is a

parallel copy of Li obtained from Li by slightly pushing off by one direction of the framing on Li .

We denote by c′′i the resulting framed sphere ci#Li . We assume that c′′i \ ci is included in Nd1∪d2

and agrees with ci outside Nd1∪d2 . The link c′′1 ∪ c′′2 is obtained from c1 ∪ c2 by component-wise

connect-summing Hopf links in Nd1∪d2 , which is realized by handle slides. Thus

(WK∪L, ∂tWK∪L, c1 ∪ c2) and (WK∪L, ∂tWK∪L, c′′1 ∪ c′′2)

are concordant.

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FAMILIES OF DIFFEOMORPHISMS AND CONCORDANCES 13

We need to show that (c′′1 ∪ c′′2) ∪ (K ∪ L) is isotopic in Nd1∪d2 to (c′1 ∪ c′2) ∪ (K ′ ∪ L′). Since

c′′1 is disjoint from d1 , the component K can be shrinked along d1 to a small sphere K ′ in a small

d-disk b around the point d1 ∩ L without intersecting c′′1 ∪ c′′2 as in Figure 10. Similarly, since c′′2

is disjoint from d2 , the component L can be shrinked along d2 to a small sphere L′ in b , without

intersecting c′′1 ∪ c′′2 , so that K ′ ∪ L′ is a small Hopf link in b .

Then similar isotopy can be performed for c′′1 ∪ c′′2 in Nd1∪d2 so that the part parallel to L1

and L2 is shrinked to a small Hopf link with bands. We may assume that this isotopy is disjoint

from b . The result of the deformation is (c′1 ∪ c′2) ∪ (K ′ ∪ L′). Thus, the deformations performed

so far give a desired concordance.

Figure 10. Sliding c1 along γ1 , and then isotoping K to K ′ .

Remark 3.7. The framed Hopf link K ∪ L may be replaced by some “smooth family” of framed

Hopf links. More precisely, let Ks ∪Ls be a smooth family of framed Hopf links parametrized over

a compact connected manifold B with a base point s0 , such that

(a) Ls = L , and hence Ls bounds d2 .

(b) d2 intersects 1-dimensional arc in c2 transversally by one point.

(c) Ks bounds a smooth family of disks d1,s in IntX such that for each s , Ls intersects d1,s

transversally by one point, and Ks intersects d2 transversally by one point.

(d) d1,s and d1,s0 agree on a neighborhood of the arc d1,s0 ∩ d2 .

Then surgery on the family Ks ∪ Ls gives a family of cobordisms that is concordant (in the sense

of Definition 3.11) to the strictly trivial family of cobordisms with a nontrivial family of spheres

c′2,s on the top, which is obtained from c2 by connected-summing with parallel copies of K1,s .

For example, if B = Sd−3 , then the family Ks may be chosen so that the associated map

D2 × B → IntX × B for the spanning disks d1,s intersects c1 × B transversally by one point in

IntX ×B . Such a family of framed Hopf links surgery will play important role in the framed link

description of the Type II surgery in Lemma 3.13.

3.4. Type I Y-surgery for links. We say that a leaf ` of a Y-graph T is simple relative to a

submanifold c in IntX with dim `+ dim c = d− 1, if the following conditions are satisfied.

(1) The leaf ` bounds a disk m in IntX .

(2) The disk m intersects c transversally by one point.

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14 BORIS BOTVINNIK AND TADAYUKI WATANABE

See Figure 11 (a). We say that a Y-graph T with leaves `1, `2, `3 is simple relative to a three

component link c1 ∪ c2 ∪ c3 in IntX with dim `i + dim ci = d − 1, if the following conditions are

satisfied.

(1) The leaves `1, `2, `3 bound disjoint disks m1,m2,m3 in IntX , respectively.

(2) For each i , the disk mi intersects ci transversally by one point and does not intersect other

components in c1 ∪ c2 ∪ c3 .

See Figure 11 (b). In this case, we take a small closed neighborhood of T ∪m1 ∪m2 ∪m3 that is

a d-disk and denote it by N(T ).

Figure 11.

Let LT be the framed link associated to T , as in Definition 3.1. We define WT as WLTin the

sense of section 3.3.1.

Lemma 3.8 (Type I surgery). Suppose that the leaves of a Y-graph T of Type I in IntX of

dimensions 1, 1, d − 2 are linked to framed submanifolds c1, c2, c3 of dimensions d − 2, d − 2, 1,

respectively, and that T is simple relative to c1 ∪ c2 ∪ c3 . Let c′1 ∪ c′2 ∪ c′3 be a framed link that is

obtained from c1 ∪ c2 ∪ c3 by component-wise connect-summing Borromean rings in N(T ). Then

the following hold.

(1) There are three disjoint small Hopf links h1, h2, h3 in N(T )\(c′1∪c′2∪c′3) and a concordance

q : W → [0, 1] between the triples

(WT , ∂tWT , c1 ∪ c2 ∪ c3) and (Wh1∪h2∪h3 , ∂tWh1∪h2∪h3, c′1 ∪ c′2 ∪ c′3)

such that the restriction of q to ((X \ IntN(T ))× 1)× [0, 1] is trivial.

(2) Moreover, if we consider up to isotopy, we may assume that two of the components of

c′1 ∪ c′2 ∪ c′3 agree as subsets of IntX with those of c1 ∪ c2 ∪ c3 .

Remark 3.9. Lemma 3.8 shows that the two definitions of Type I surgeries: “the complement of

thickened string link” given in section 2, and “framed link surgery” in Definition 3.1 are equivalent.

Namely, let L be the six component framed link of Definition 3.1 in V . Then the latter definition

is given by surgery along L in V . According to Lemma 3.8 and if we consider modulo small Hopf

links, this surgery replaces V with another one that is obtained by taking the complements of the

Borromean string link. The relative diffeomorphism type of the resulting manifold is determined

uniquely and agrees with the former definition of Type I surgery.

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FAMILIES OF DIFFEOMORPHISMS AND CONCORDANCES 15

Figure 12.

Proof of Lemma 3.8. Lemma 3.8 is obtained by iterated applications of the concordance deforma-

tions of Lemma 3.6. Namely, by Definition 3.1, the surgery on T is given by the surgery on a six

component framed link⋃3i=1(Ki ∪ Li). Since T is simple relative to c1 ∪ c2 ∪ c3 , the component

Ki bounds a disk di in N(T ). After relabeling if necessary, we may assume that for each i , the

intersections di ∩ ci and di ∩ Li are both one point and orthogonal, and di does not have other

intersections with other link components. (See Figure 12 (d).) We choose an embedded path γi in

di that goes from di ∩ ci to di ∩Li . Then we may define the band sum ci#Li along γi so that the

result is disjoint from di , as in the proof of Lemma 3.6.

After performing surgeries on the framed link⋃3i=1(Ki ∪ Li), the component ci can be slid

over γi and then over the (dimLi + 1)-handle attached to Li . The result of the handle slide is

ci#Li , where Li is a parallel copy of Li obtained from Li by slightly pushing off by one direction

of the framing on Li . We define c′i as the resulting framed sphere ci#Li . We assume that Li is

included in N(T ) and that c′i agrees with ci outside N(T ). Now a framed link c′1 ∪ c′2 ∪ c′3 has

been obtained from c1 ∪ c2 ∪ c3 by sliding components over the handles attached to⋃3i=1(Ki ∪Li),

and also can be obtained by component-wise connect-summing Borromean rings in N(T ). (See

Figure 12 (e).)

We need to show that the Hopf links Ki ∪Li can be deformed into a small Hopf link hi . Since

c′1 is disjoint from d1 , the component K1 can be shrinked along d1 to a small sphere K ′1 in a small

d-disk around the point d1 ∩ L1 , without intersecting c′1 ∪ c′2 ∪ c′3 during the shrinking isotopy.

Then by sliding other components Kj ∪ Lj over K ′1 for j 6= 1, the component L1 can be made

unlinked from Kj ∪ Lj . This slide does not change the isotopy type of

(K2 ∪ L2) ∪ (K3 ∪ L3) ∪ c′1 ∪ c′2 ∪ c′3

in N(T ), though does change that in N(T )\(K ′1∪L1). Now the Hopf link K ′1∪L1 can be shrinked

into a small Hopf link h1 without affecting other components. After that, similar slidings can be

performed for the Hopf links K2 ∪ L2 and K3 ∪ L3 so that they can be separated and shrinked

into disjoint small Hopf links h2, h3 , respectively. Thus the deformations performed so far consist

of isotopy and slides over handles, which give a desired concordance as in (1). The condition (2)

follows from Property 2.1.

The following lemma is an analogue of Habiro’s move 10 ([Ha, Proposition 2.7]).

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16 BORIS BOTVINNIK AND TADAYUKI WATANABE

Lemma 3.10 (Type I with Null-leaf). Suppose that the leaves `1, `2, `3 of a Y-graph T of Type

I in IntX bound disjoint disks m1,m2,m3 in IntX , respectively. Suppose there are disjoint sub-

manifolds c1, c2 in IntX \ T such that `i is simple relative to ci for i = 1, 2, and that c1 ∪ c2

is disjoint from m3 . (See Figure 13.) Then there are three disjoint small Hopf links h1, h2, h3 in

N(T ) \ (c1 ∪ c2) and a concordance q : W → [0, 1] between the triples

(WT , ∂tWT , c1 ∪ c2) and (Wh1∪h2∪h3 , ∂tWh1∪h2∪h3, c1 ∪ c2)

such that the restriction of q to ((X \ IntN(T ))× 1)× [0, 1] is trivial.

Figure 13. Y-graph with null-leaf

Proof. This is a corollary of Lemma 3.8. It suffices to delete c3 and c′3 in Lemma 3.8. By Prop-

erty 2.1 of Borromean rings, (c′1 ∪ c′2) ∩N(T ) in Lemma 3.8 is isotopic to (c1 ∪ c2) ∩N(T ) fixing

the boundary.

3.5. Type II Y-surgery for links. We shall give analogues of Lemmas 3.8 and 3.10 for Type II

Y-surgery, which are Lemmas 3.13 and 3.15.

3.5.1. Concordance of a family of cobordisms.

Definition 3.11. (1) A concordance of a (W,∂tW )-bundle over B is a (W,∂tW )-bundle over

B× [0, 1] whose restriction over B×0 is identified with the given (W,∂tW )-bundle. We

say that (W,∂tW )-bundles pi : Ei → B , i = 0, 1, are concordant if there is a concordance

q : E → B× I of p0 such that q−1(B×i) = Ei and q|B×i = pi under the identification

B × i = B .

(2) For fiberwise framed trivialized subbundles L and L′ of ∂+E0 and ∂+E1 , respectively,

a concordance between (p0, L) and (p1, L′) is a concordance q : E → B × I between p0

and p1 that has a trivialized subbundle C of ∂+E with a fiberwise framing such that it

restricts to the framed trivialized subbundles L and L′ on q−1(B×0) and q−1(B×1),

respectively.

(3) For (p0, L) as above, we define pL0 : EL0 → B to be the (WL, ∂tW )-bundle obtained by

attaching families of handles along L , where L is a fiber of L .

The following lemma is evident from definition.

Lemma 3.12. Let pi : Ei → B , i = 0, 1, be (W,∂tW )-bundles as in Definition 3.11.

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FAMILIES OF DIFFEOMORPHISMS AND CONCORDANCES 17

(1) A concordance between p0 and p1 induces a relative bundle isomorphism between them.

(2) A concordance between (p0, L) and (p1, L′) induces a relative bundle isomorphism between

p0 and p1 that restricts to a bundle isomorphism between L and L′ .

(3) A concordance between (p0, L) and (p1, L′) induces a concordance between pL0 and pL

′1 .

3.5.2. Type II Y-surgery.

Lemma 3.13 (Type II surgery). Suppose that the leaves of a Y-graph T of Type II in IntX of

dimensions 1, d − 2, d − 2 are linked to framed submanifolds c1, c2, c3 of dimensions d − 2, 1, 1,

respectively, and that T is simple relative to c1 ∪ c2 ∪ c3 . Let c′1 ∪ c′2 ∪ c′3 be a framed trivialized

subbundle of IntX × Sd−3 → Sd−3 that is obtained from (c1 ∪ c2 ∪ c3)× Sd−3 → IntX × Sd−3 by

fiberwise component-wise connect-summing Sd−3 -family of framed links Sd−2 ∪ S1 ∪ S1 → N(T )

that defines the Type II surgery. Then the following hold.

(1) There is a concordance q : E → Sd−3 × [0, 1] between the pairs of bundles over Sd−3

(pT0 , (c1 ∪ c2 ∪ c3)× Sd−3) and (ph1∪h2∪h30 , c′1 ∪ c′2 ∪ c′3)

such that the restriction of q to ((X \ IntN(T ))× Sd−3 × 1 is trivial.

(2) We may assume that two of the components of c′1 ∪ c′2 ∪ c′3 agree with those of

(c1 ∪ c2 ∪ c3)× Sd−3 → IntX × Sd−3.

Remark 3.14. Lemma 3.13 shows that the two definitions of Type II surgeries given in section 2

and Definition 3.2 are equivalent, as in Remark 3.9.

Proof of Lemma 3.13. Proof is analogous to that of Lemma 3.8. Lemma 3.13 is obtained by iter-

ated applications of the concordance deformations of Remark 3.7. We only need to replace Li in

Lemma 3.8 with a family of links Li,s in N(T ), s ∈ Sd−3 .

By Definition 3.2, the surgery on T is given by the surgery on a six component link⋃3i=1(Ki ∪ Li,s)

in each fiber over s ∈ Sd−3 . We assume that for all s , Li,s agrees with Li,s0 near the base point

of Li,s0 . Then Ki ∪ Li,s satisfies the conditions (a)–(d) of Remark 3.7.

Since T is simple relative to c1 ∪ c2 ∪ c3 , the component Ki bounds a disk di in N(T ). After

relabeling if necessary, we may assume that for each i , the intersections di∩ci and di∩Li,s are both

one point and orthogonal, and di does not have other intersections with other link components.

Moreover, we assume that di ∩ Li,s consists of the base point of Li,s , which agrees with that of

Li,s0 . We choose an embedded path γi in di that goes from di∩ci to the base point di∩Li,s0 . Then

we may connect-sum ci with Li,s along γi so that the result is disjoint from di , as in the proof

of Lemma 3.8. The fiberwise connected sum produces a smooth family with respect to s ∈ Sd−3

since γi is connected to the base point of Li,s near which Li,s agrees with Li,s0 by assumption.

This procedure defines the band sum ci#Li,s .

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18 BORIS BOTVINNIK AND TADAYUKI WATANABE

After performing surgeries on the framed link⋃3i=1(Ki ∪ Li,s), the component ci can be slid

over γi and then over the (dimLi,s + 1)-handle attached to Li,s . The result of the handle slide

is ci#L′i,s , where L′i,s is a parallel copy of Li,s obtained from Li,s by slightly pushing off by one

direction of the framing on Li,s . We define c′i,s as the resulting framed sphere ci#L′i,s . We assume

that L′i,s is included in N(T ) and that c′i,s agrees with ci outside N(T ). Let b1,s ∪ b2,s ∪ b3,s be

the Sd−3 -family of framed links Sd−2 ∪ S1 ∪ S1 → N(T ) that defines the Type II surgery, with

dim b1,s = d − 2, dim b2,s = dim b3,s = 1. Now a framed link c′1,s ∪ c′2,s ∪ c′3,s has been obtained

from c1 ∪ c2 ∪ c3 by sliding components over the handles attached to⋃3i=1(Ki ∪Li,s), and also can

be obtained by component-wise connect-summing b1,s ∪ b2,s ∪ b3,s in N(T ). The family of framed

links c′1,s ∪ c′2,s ∪ c′3,s can be defined smoothly with respect to s ∈ Sd−3 as a trivialized framily.

We need to show that the family of Hopf links Ki∪Li,s can be deformed into a small Hopf link

hi for all s simultaneously. Since c′1,s is disjoint from d1 , the component K1 can be shrinked along

d1 to a small sphere K ′1 in a small d-disk around the (base) point d1 ∩ L1,s , without intersecting

c′1,s ∪ c′2,s ∪ c′3,s during the shrinking isotopy. Then by sliding other components Kj ∪Lj,s over K ′1

for j 6= 1, the component L1,s can be made unlinked from Kj ∪ Lj,s . This slide does not change

the fiber isotopy type of

(K2 ∪ L2,s) ∪ (K3 ∪ L3,s) ∪ c′1,s ∪ c′2,s ∪ c′3,s

in N(T ), though does change that in N(T ) \ (K ′1 ∪ L1,s). Now the Hopf link K ′1 ∪ L1,s can be

shrinked into a small Hopf link h1 simultaneously for all s without affecting other components.

After that, similar slidings can be performed for the Hopf links K2 ∪ L2,s and K3 ∪ L3,s so that

they can be separated and shrinked into disjoint small Hopf links h2, h3 , respectively, by a fiberwise

isotopy. Thus the deformations performed so far consist of fiberwise isotopy/slides over handles,

which give a desired concordance as in (1). The condition (2) follows from Property 2.1 again.

Lemma 3.15 (Type II with Null-leaf). Suppose that the leaves `1, `2, `3 of a Y-graph T of Type II

in IntX bound disjoint disks m1,m2,m3 in IntX , respectively. Suppose there are disjoint framed

submanifolds c1, c2 in IntX \ T such that `i is simple relative to ci for i = 1, 2, and that c1 ∪ c2

is disjoint from m3 . Then if we consider modulo small Hopf links and concordance, surgery on T

does not change (c1 ∪ c2)× Sd−3 .

Proof. This is a corollary of Lemma 3.13. It suffices to delete c3 and c′3,s in Lemma 3.13. By

Property 2.1 of Borromean rings, (c′1,s ∪ c′2,s)∩N(T ) in Lemma 3.13 is isotopic to (c1 ∪ c2)∩N(T )

fixing the boundary.

4. Family of framed links for graph surgery

In this section, we shall nearly complete the proof of Theorem 2.3 (1), by proving the corre-

sponding statement for BΓ -family instead of Sk(d−3) -family.

Proposition 4.1. Let Γ be a labeled edge-oriented trivalent graph as in section 2 with 2k vertices.

The (X, ∂)-bundle πΓ : EΓ → BΓ for an embedding φ : Γ→ IntX is concordant to a (X, ∂)-bundle

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FAMILIES OF DIFFEOMORPHISMS AND CONCORDANCES 19

obtained from the product bundle X × Sk(d−3) → Sk(d−3) by fiberwise surgeries along a BΓ -family

of framed links hs : S1 ∪ Sd−2 → IntX , s ∈ BΓ , that satisfies the following conditions:

(1) hs is isotopic to the Hopf link for each s.

(2) The restriction of hs to Sd−2 component is a constant BΓ -family.

(3) There is a small neighborhood N of Imφ such that the image of hs is included in N for

all s ∈ BΓ .

We will prove this by trying to construct a concordance between the families of cobordisms for

the two surgeries and by restriction to the top faces. Of course, there is no such concordance in

the obvious sense since the numbers of components of the framed links for Γ-surgery and surgery

along a family of Hopf-links are different. We modify the assumption slightly so that a concordance

between the two families of cobordisms will make sense.

Let b be a small d-disk and let w be the relative cobordism obtained from b × I by surgery

along a small Hopf link h in Int b×1 . For a relative cobordism W between ∂−W = X×0 and

∂+W ∼= X such that ∂W = ∂+W ∪∂X×1 (∂X × I) ∪∂X×0 ∂−W , let WN denote the boundary

connected sum of W and N copies of w along disjoint union of disks Dd−1 × I ⊂ ∂X × I .

Figure 14. The cobordism WN

Now we set W = X × I and let pΓ : WΓ → BΓ be the (W6k, ∂tW6k)-bundle obtained from

the trivial W -bundle by surgery along the associated family of (6k × 2 = 12k component) framed

links in X × 1 for the Γ-surgery. The restriction of this bundle to the top face gives the former

(X, ∂)-bundle πΓ : EΓ → BΓ of Proposition 4.1. The number 6k is because there are 2k Y-graphs

for the Γ-surgery each gives rise to 3 Hopf links. On the other hand, the latter (X, ∂)-bundle of

Proposition 4.1 is the top face of a (W1, ∂tW1)-bundle over BΓ .

We add to WΓ one more Hopf link surgery without changing the (X, ∂)-bundle on the top

face, as follows. Let G1 ∪ · · · ∪ G2k be the Y-link for the embedding φ of Γ. Let a1 ∪ b1 be the

framed Hopf link for the first edge of Γ as in Figure 3, which are leaves of some Y-graphs. We

replace a1 ∪ b1 by a framed “Hopf chain” c1 ∪ c2 ∪ c3 ∪ c4 such that

• dim c1 = dim c3 = dim a1 , dim c2 = dim c4 = dim b1 ,

• ci ∪ ci+1 is a Hopf link for i = 1, 2, 3.

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20 BORIS BOTVINNIK AND TADAYUKI WATANABE

Figure 15. The Hopf chain c1 ∪ c2 ∪ c3 ∪ c4

Then the leaves a1 and b1 are replaced by c1 and c4 , respectively, and G1 ∪ · · · ∪ G2k becomes

a Y-link G′1 ∪ · · · ∪ G′2k that is linked to the Hopf link c2 ∪ c3 . The Y-link G′1 ∪ · · · ∪ G′2k is the

one obtained from a uni-trivalent graph G attached to the link L = c2 ∪ c3 , as in Proposition 4.3

below. By Lemma 3.6, this replacement does not change the concordance class of the triple up to

small Hopf links. Namely, there are a small Hopf link h in ∂+W that is disjoint from the Y-link

G1 ∪ · · · ∪G2k and L , and a concordance between the triples

(WΓ, ∂tWΓ, h×BΓ) and (WG′1∪···∪G′2k , ∂tW

G′1∪···∪G′2k , (c2 ∪ c3)×BΓ),

where h corresponds to c2 ∪ c3 . Let pΓ1 : WΓ

1 → BΓ be the (W6k+1, ∂tW6k+1)-bundle given by

fiberwise surgery

(WG′1∪···∪G′2k , ∂tWG′1∪···∪G′2k)(c2∪c3)×BΓ .

The number 6k+ 1 is due to the addition of c2 ∪ c3 . The newly added Hopf link c2 ∪ c3 will serve

as the family of Hopf links hs of Proposition 4.1.

Proposition 4.1 is an immediate corollary of the following lemma, which gives an extension of

Proposition 4.1 to cobordisms.

Lemma 4.2. Let Γ be as in Proposition 4.1. The above (W6k+1, ∂tW6k+1)-bundle pΓ1 : WΓ

1 → BΓ

determined by an embedding φ : Γ→ IntX×1 is concordant to a (W6k+1, ∂tW6k+1)-bundle that

is obtained from the product W -bundle W ×BΓ → BΓ by fiberwise handle attachments along some

BΓ -family of framed links hs : S1∪Sd−2 → IntX×1, s ∈ BΓ , and fiberwise boundary connected

sums with 6k copies of the trivial (w, ∂tw)-bundle p0 : e → BΓ , e = w × BΓ , where hs satisfies

the conditions (1), (2), (3) of Proposition 4.1.

Lemma 4.2 will follow as a special case of a more general result, which is stated in Proposi-

tion 4.3 below, generalized to a uni-trivalent graph attached to a link. To state the general result,

let us make some assumptions. Let G be a connected uni-trivalent graph embedded in IntDd such

that

(1) G has r trivalent vertices and at least one univalent vertex,

(2) edges are oriented in a way that the orientations of edges at each trivalent vertex is the

same as that of Y-graph of Type I or II,

(3) the univalent vertices of G are on components of some spherical link L in IntDd consisting

of 1- and (d− 2)-spheres,

(4) L ∩ IntG = ∅ , where IntG is the complement of the union of univalent vertices in G ,

(5) each univalent vertex of G that is “inward” to G is attached to a (d− 2)-sphere in L ,

(6) each univalent vertex of G that is “outward” from G is attached to a 1-sphere in L .

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FAMILIES OF DIFFEOMORPHISMS AND CONCORDANCES 21

Figure 16. A uni-trivalent graph G attached to a link L

We take a small closed neighborhood N(G) of G such that its intersection with L consists of 1-

and (d− 2)-disks each of which is a small neighborhood of a univalent vertex of G in a component

of L . As before, we may construct a Y-link G1 ∪ · · · ∪Gr inside N(G) by putting a framed Hopf

link at each edge of G between trivalent vertices and by replacing each univalent vertex with a leaf

that bounds a disk in N(G) transversally intersecting L at a point. We call such a leaf a simple

leaf of G relative to L . Then we define surgery on G by the surgery on the Y-link G1 ∪ · · · ∪Gr .

Let BG = Sa1 × Sa2 × · · · × Sar , where ai = 0 or d− 3 depending on whether Gi is of Type

I or II, respectively. Let pG : WG → BG be the (W3r, ∂tW3r)-bundle obtained from the trivial

(X × I)-bundle over BG by surgery along the associated set of families of framed Hopf links in

X × 1 for G as in Definitions 3.1 and 3.2.

Proposition 4.3. Let G be a connected uni-trivalent graph embedded in IntX attached to some

link L as above. Let b1, . . . , b3r be disjoint small d-balls in N(G) \ (L ∪ G1 ∪ · · · ∪ Gr) and let

h1, . . . , h3r be small Hopf links in N(G)\ (L∪G1∪· · ·∪Gr) such that hi ⊂ Int bi . (See Figure 17.)

Let LN(G) = L∩N(G). Then there is a BG -family of embeddings of the union of disks LN(G) into

N(G) \ (b1 ∪ · · · ∪ b3r):

Φs : LN(G) → N(G) \ (b1 ∪ · · · ∪ b3r) (s ∈ BG)

that agree with the inclusion near ∂N(G) such that there is a concordance between the triples

(WG, ∂tWG, L×BG) and (W h1∪···∪h3r ×BG, ∂tW h1∪···∪h3r ×BG, L),

where L =⋃s∈BG

Ls , which is a trivialized subbundle of ∂+W3r × BG = X × BG , and Ls is

obtained from L by replacing LN(G) by Φs . Moreover, for any choice of a component ` of LN(G) ,

we may assume that the restriction of Φs to all the components in LN(G) \ ` does not depend on

the parameter s, after a fiberwise isotopy, which depends on the choice of `.

Proof. We prove this by induction on r . The case r = 1 has been proved in Lemma 3.8 or 3.13.

We next consider the case r = 2 as a warm-up case, though this case is included in the next step.

When r = 2, there are two possibilities for G satisfying the assumption of Proposition 4.3:

(a) a connected uni-trivalent tree with four legs (Figure 18, left), or

(b) a loop with two legs attached (Figure 18, right).

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22 BORIS BOTVINNIK AND TADAYUKI WATANABE

Figure 17. N(G), L , bi

Figure 18.

First, we consider the effect of the surgery on G2 . Roughly, it gives an Sa2 -family of embeddings

of the Y-graph G1 in N(G). Let `1j be a leaf of G1 that is not simple relative to L (Figure 11 (a)

for a simple leaf relative to L). Let L(2) be the union of L and all such leaves `1j . We deform the

triple (WG2 , ∂tWG2 , L(2) × Sa2) by Lemma 3.8 or 3.13 so that L is fixed over Sa2 . Namely, let

δ2 = h21 ∪ h2

2 ∪ h23 be the union of three disjoint small Hopf links in N(G). By Lemma 3.8 or 3.13,

we see that there are Sa2 -families of leaves ˜1j =⋃s2∈Sa2 `

1j,s2

in N(G) and a concordance between

(WG2 , ∂tWG2 , L(2)× Sa2) and (W δ2 × Sa2 , ∂tW

δ2 × Sa2 , L(2)), where L(2) =⋃j˜1j ∪ (L× Sa2),

such that for each s2 , `1j,s2 agrees with `1j near the base point of the leaf, where an edge of G1 is

attached. Since N(G2) is disjoint from the Y-shaped part of G1 , the result extends to a family of

embeddings of G1 in N(G) \ L :

ϕs2 : G1 → N(G) \ L (s2 ∈ Sa2).

Each simple leaf `1i of G1 relative to L bounds a disk m1i in N(G). We take m1

i for each such i

and take a small closed neighborhood of G1∪⋃im

1i in N(G) and denote it by N0(G1). By isotopy

extension, the family ϕs2 can be extended to a family of embeddings ϕ′s2 : N0(G1) → N(G)

(s2 ∈ Sa2 ).

Next, we consider the effect of the surgery on G1 . Let `2j be a leaf of G2 that is not sim-

ple relative to L . Let L(1) be the union of L and all such leaves `2j . We deform the triple

(WG1 , ∂tWG1 , L(1) × Sa1) by concordance of Lemma 3.8 or 3.13 so that `2j is fixed over Sa1 .

Namely, let δ1 = h11 ∪ h1

2 ∪ h13 be the union of three disjoint small Hopf links in N(G) \ δ2 . By

Lemma 3.8 or 3.13, we see that there are an Sa1 -family of disks from LN(G) inside N0(G1):

ϕs1 : D → N0(G1) (s1 ∈ Sa1),

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FAMILIES OF DIFFEOMORPHISMS AND CONCORDANCES 23

where D = L ∩ N0(G1), and a concordance between (WG1 , ∂tWG1 , L(1) × Sa1) and (W δ1 ×

Sa1 , ∂tWδ1 × Sa1 , L(1)), where L(1) =

⋃j(`

2j × Sa1)∪

⋃s1∈Sa1 Ls1 and Ls1 is obtained from L by

replacing D by the family of its embeddings ϕs1 . By the last sentence of Lemma 3.8 or 3.13, we

may assume that only one component of D may depend on s1 .

Now we combine the two surgeries for G1 and G2 . Let L be the trivialized subbundle of

X×BG obtained from L×BG by replacing its intersection with N0(G1)×BG by the composition

ϕ′s2 ϕs1 : D → N(G) (s1, s2) ∈ Sa1 × Sa2 = BG.

By the results of the previous paragraphs, we see that there is a concordance between the triples

(WG, ∂tWG, L × BG) and (W δ1∪δ2 × BG, ∂tW δ1∪δ2 × BG, L). Note that the restriction of L to

the components of LN(G) that intersect N(Gi) (i = 1, 2) is a strictly trivial family except one

component. This completes the proof for r = 2.

For general r , it suffices to replace G1 in the r = 2 case with Gr , and G2 with a Y-link

G1 ∪ · · · ∪Gr−1 corresponding to a connected uni-trivalent graph G′ with r − 1 trivalent vertices.

First, we assume that the result holds true for G(r−1) = G1∪· · ·∪Gr−1 . We only need a special

case of this. Namely, let `rj be a leaf of Gr that is not simple relative to L (Figure 11 (a) for a

simple leaf relative to L). Let L(1, . . . , r− 1) be the union of L and all such leaves `rj . Let δ(r−1)

be the union of 3(r − 1) disjoint small Hopf links in N(G). By assumption, we see that there are

BG′ -family of disks from LN(G) inside N0(G(r−1)):

ϕs1,...,sr−1 : D → N0(G(r−1)) ((s1, . . . , sr−1) ∈ BG′),

where D = L(1, . . . , r − 1) ∩N0(G(r−1)), and a concordance between

(WG(r−1) , ∂tWG(r−1) , L(1, . . . , r − 1)×BG′) and

(W δ(r−1) ×BG′ , ∂tW δ(r−1) ×BG′ , L(1, . . . , r − 1)),

where L(1, . . . , r − 1) =⋃j(`

rj × Sar) ∪

⋃(s1,...,sr−1)∈BG′

Ls1,...,sr−1 and Ls1,...,sr−1 is obtained from

L by replacing D by ϕs1,...,sr−1 . The restriction of ϕs1,...,sr−1 to the components of D intersecting

N(Gi) (1 ≤ i ≤ r − 1) except one, is a strictly trivial family.

We consider the effect of the surgery on Gr . Let `(r−1)j be a leaf of G1 ∪ · · · ∪ Gr−1 that

corresponds to a univalent vertex of G′ and that is not simple relative to L . Let L(r) be the union

of L and all such leaves `(r−1)j . Let δr = hr1∪hr2∪hr3 be the union of three disjoint small Hopf links

in N(G). By Lemma 3.8 or 3.13, we see that there are Sar -families of leaves ˜(r−1)j =

⋃sr∈Sar `

(r−1)j,sr

in N(G) and a concordance between

(WGr , ∂tWGr , L(r)× Sar) and (W δr × Sar , ∂tW δr × Sar , L(r)),

where L(r) =⋃j˜(r−1)j ∪ (L× Sar), such that for each sr , `

(r−1)j,sr

agrees with `(r−1)j near the base

point. This gives a family of embeddings of G(r−1) in N(G) \ L :

ϕsr : G(r−1) → N(G) \ L (sr ∈ Sar).

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24 BORIS BOTVINNIK AND TADAYUKI WATANABE

By isotopy extension, the family ϕsr can be extended to a family of embeddings ϕ′sr : N0(G(r−1))→N(G) (sr ∈ Sar ).

Now we combine the two surgeries for G(r−1) and Gr . Let L be the trivialized subbundle of

X ×BG obtained from L×BG by replacing LN0(G(r−1)) ×BG by the composition

ϕ′sr ϕs1,...,sr−1 : D → N(G) (s1, . . . , sr) ∈ BG.

By the results of the previous paragraphs, we see that there is a concordance between the triples

(WG, ∂tWG, L × BG) and (W δ(r−1)∪δr × BG, ∂tW δ(r−1)∪δr × BG, L). Note that the restriction of

L to the components of LN(G) that intersect N(Gi) (1 ≤ i ≤ r ) is a strictly trivial family except

one component. This completes the induction.

Proof of Lemma 4.2. We assume without loss of generality that dim c2 = 1 and dim c3 = d − 2.

Applying Proposition 4.3 for the Y-link G′1 ∪ · · · ∪ G′2k and L = c2 ∪ c3 , we see that surgery on

G′1 ∪ · · · ∪ G′2k produces a BΓ -family of embeddings of LN(G) into N(G), whose restriction to

c3 ∩N(G) is a trivial family. This gives the desired family of framed Hopf links.

5. Bordism modification to a Sk(d−3) -family of surgeries

5.1. From a BΓ -family to a Sk(d−3) -family. We shall complete the proof of Theorem 2.3 (1).

Proposition 5.1. Let G be a uni-trivalent graph attached to a framed link L, as in Proposition 4.3.

The BG = Sa1 × · · · × Sar -family of framed embeddings of disks LN(G) = L ∩ N(G) in N(G) of

Proposition 4.3 can be deformed into an Sa1+···+ar -family by an oriented bordism in the space

Embfr∂ (LN(G), N(G)).

To prove Proposition 5.1, we shall instead prove the following stronger lemma.

Lemma 5.2. The map BG → Embfr∂ (LN(G), N(G)) for the BG -family of Proposition 4.3 factors

up to homotopy over a map BG → Sa1+···+ar of degree 1.

Proof. We prove this by induction on r . The case r = 1 is obvious. Assume that the map

gr−1 : BG′ = Sa1 × · · · × Sar−1 → Embfr∂ (LN(G), N(G′))

for a Y-link G1 ∪ · · · ∪Gr−1 that corresponds to a connected uni-trivalent graph G′ factors up to

homotopy into a degree 1 map Sa1 × · · · × Sar−1 → Sa1+···+ar−1 and a map

gr−1 : Sa1+···+ar−1 → Embfr∂ (LN(G′), N(G′)).

Since gr−1 is null-homotopic, one may apply Lemma 5.3 below, and the map gr−1 is null-homotopic.

Adding one more Y-graph Gr so that G1 ∪ · · · ∪Gr corresponds to a connected uni-trivalent

graph G , we obtain a map gr : BG′ × Sar → Embfr∂ (LN(G), N(G)) that factors up to homotopy

over a degree 1 map BG′ × Sar → Sa1+···+ar−1 × Sar .

The restrictions of the induced map

gr : Sa1+···+ar−1 × Sar → Embfr∂ (LN(G), N(G))

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FAMILIES OF DIFFEOMORPHISMS AND CONCORDANCES 25

to the subspaces Sa1+···+ar−1 ×∗ and ∗×Sar of Sa1+···+ar−1 ×Sar are pointed null-homotopic

in Embfr∂ (LN(G), N(G)) by Lemma 3.10 or 3.15 and by the nullity of gr−1 in

Embfr∂ (LN(G′), N(G′)).

Thus the map gr factors up to homotopy over a degree 1 map Sa1+···+ar−1 ×Sar → Sa1+···+ar .

Lemma 5.3 ([Wa18a, Proof of Lemma B]). Let B = Sa1 × Sa2 × · · · × Sas and let

A =

s⋃i=1

s∏j=1

Qij

= (∗ × Sa2 × · · · × Sas) ∪ (Sa1 × ∗ × · · · × Sas) ∪ · · · ∪ (Sa1 × Sa2 × · · · × ∗),

where Qij = Saj if j 6= i, and Qij = ∗ ⊂ Saj if j = i. For a space Y , suppose that we

have a pointed null-homotopy of a pointed map g : B → Y and another pointed null-homotopy

of the restriction g|A : A → Y . Then g can be factored up to homotopy into a pointed map

B → B/A ' Sa1+···as and a null-homotopic map B/A→ Y .

Proof. First, note that B/A ' B ∪A CA and ΣA ' (B ∪A CA) ∪B CB , and recall the long exact

sequence of sets of homotopy classes of pointed maps

[ΣB, Y ]→ [ΣA, Y ]→ [B/A, Y ]→ [B, Y ]→ [A, Y ].

For our choice of B and A , the natural map ΣA→ ΣB induced by the inclusion splits with cofiber

Σ(B/A) ([BBCG, p.1662]), and the map [ΣA, Y ]→ [B/A, Y ] in the above exact sequence is trivial.

On the other hand, the map g can be extended to a map g′ : B ∪ACA→ Y by the null-homotopy

of g|A . Moreover, this can be extended to a map g′′ : (B∪ACA)∪BCB → Y by the null-homotopy

of g . Hence the class of g′ in [B/A, Y ] is trivial. We have the factorization of g

B → B/A→ Y

up to homotopy, where the latter map is pointed null-homotopic.

Corollary 5.4. The BΓ -family of framed links hs : S1∪Sd−2 → IntX , s ∈ BΓ , in Proposition 4.1

can be deformed by a bordism in the space of embeddins, into a Sk(d−3) -family of framed embeddings

S1 ∪ Sd−2 → IntX that satisfies the following conditions:

(1) hs is isotopic to the Hopf link for each s.

(2) The restriction of hs to Sd−2 component is a constant Sk(d−3) -family.

(3) There is a small neighborhood N of Imφ such that the image of hs is included in N for

all s ∈ Sk(d−3) .

Hence, fiberwise handle attachments along the family of embeddings over the bordism gives a bundle

bordism of cobordism bundles pΓ1 : WΓ

1 → BΓ to a (W6k+1, ∂tW6k+1)-bundle over Sk(d−3) , which

restricts on the top face to a (X, ∂)-bundle bordism between πΓ : EΓ → BΓ and a (X, ∂)-bundle

$Γ : EΓ → Sk(d−3) .

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26 BORIS BOTVINNIK AND TADAYUKI WATANABE

5.2. Modification into a family of h-cobordisms. We prove Theorem 2.3 (2).

Proposition 5.5. There exists a (X × I)-bundle ΠΓ : WΓh → Sk(d−3) with structure group

Difft(X × I) such that

(1) the fiberwise restriction of ΠΓ to X × 1 is $Γ ,

(2) WΓh is obtained by attaching Sk(d−3) -families of 1- and 2-handles to the product X × I -

bundle (X × I)× Sk(d−3) → Sk(d−3) at (X × 1)× Sk(d−3) .

Proof. By Corollary 5.4, there is a cobordism bundle E → Sk(d−3) which is obtained from the

strictly trivial (X × I)-bundle by attaching families of 2- and (d− 1)-handles along hs and whose

restriction to ∂+E agrees with $Γ .

Let E1 be the family of handlebodies obtained by attaching only the family of (d− 1)-handles

to the strictly trivial (X × I)-bundle by hs|Sd−2 . Since the attaching map hs|Sd−2 of the family

of (d − 1)-handles is a strictly trivial family by Corollary 5.4, the family E1 is a strictly trivial

bundle, on the top of which the 2-handle may be attached along the attaching sphere induced by

hs|S1 on ∂+E1 that may not be strictly trivial.

Attaching a (d − 1)-handle to X × I along an unknotted framed (d − 2)-sphere on X × 1turns the top face into X#(Sd−1 × S1). Also, the same manifold can be obtained by attaching a

1-handle along a framed 0-sphere on X × 1 instead of a (d − 1)-handle. Thus, we may replace

the strictly trivial bundle E1 by another family E′1 of handlebodies that is obtained by attaching

strictly trivial family of 1-handles to X × I , without changing the manifold

∂+E1 = (X#(Sd−1 × S1))× Sk(d−3).

Then we attach a family of 2-handles to E′1 along the attaching spheres induced by hs|S1 on

∂+E′1 = ∂+E1 . The resulting bundle ΠΓ : WΓ

h → Sk(d−3) is a (X × I)-bundle, since the two

Figure 19.

handles are in a cancelling position in a fiber, namely, the descending disk of the 2-handle and the

ascending disk of the 1-handle intersects transversally in one point in ∂+E1 . Then by M. Morse’s

result [Mo] (see also [Mi, Theorem 5.4 (First Cancellation Theorem)]), the pair of two handles can

be eliminated and the cobordism can be modified into the trivial h-cobordism. By construction,

∂+WΓh = E

Γand πΓ restricts to $Γ .

Remark 5.6. We notice that by construction, the bundle ΠΓ : WΓh → Sk(d−3) admits a fiberwise

Morse function f : WΓh → R and a fiberwise gradient-like vector field ξ for f such that the family

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FAMILIES OF DIFFEOMORPHISMS AND CONCORDANCES 27

of handle decompositions for ξ agrees with that of the 1- and 2-handles in Proposition 5.5. Such

a family of Morse functions can be constructed by applying [Mi, Theorem 3.12] for the families of

handles, which is possible since the families of handles are given by families of attaching maps, and

the construction of the Morse function in the proof of [Mi, Theorem 3.12] for the surgery χ(V, ϕ)

depends smoothly on the attaching maps ϕ .

6. Odd dimensional case

Now we shall prove another version of Theorem 2.3 for odd dimensional disks.

Theorem 6.1. Let d = 2m + 1 ≥ 5 and let Γ be a labeled edge-oriented trivalent graph as in

section 2 with 2k vertices.

(1) The (X, ∂)-bundle πΓ : EΓ → BΓ for an embedding φ : Γ → IntX is related by a (X, ∂)-

bundle bordism to a (X, ∂)-bundle $Γ : EΓ → Sk(d−3) = S2k(m−1) obtained from the trivial

bundle over Sk(d−3) by fiberwise surgeries along a Sk(d−3) -family of framed embeddings

hs : Sm ∪ Sm → IntX , s ∈ Sk(d−3) , that satisfies the following conditions:

(a) hs is isotopic to the Hopf link for each s.

(b) The restriction of hs to one of Sm components is a constant Sk(d−3) -family.

(c) There is a small neighborhood N of Imφ such that the image of hs is included in N

for all s ∈ Sk(d−3) .

(2) There exists a (X × I)-bundle ΠΓ : WΓh → Sk(d−3) with structure group Difft(X × I) such

that

(a) the fiberwise restriction of ΠΓ to X × 1 is $Γ ,

(b) WΓh is obtained by attaching Sk(d−3) -families of (d+ 1)-dimensional m- and (m+ 1)-

handles to the product X × I -bundle (X × I) × Sk(d−3) → Sk(d−3) at the top portion

(X × 1)× Sk(d−3).

This can be proved by replacing dimensions of the framed link components everywhere in

section 3–5, as follows:

• Hopf link S1∪Sd−2 → X by Hopf link Sm∪Sm → X . Accordingly, the phrase “restriction

of hs to Sd−2 component” in Proposition 4.1 should be replaced by “restriction of hs to

one of Sm components”.

• Y-surgery is given by an Sm−1 -family of embeddings Dm ∪Dm ∪Dm → D2m+1 obtained

by parametrizing a Borromean string link D2m−1 ∪Dm ∪Dm → D2m+1. We take this at

each trivalent vertex, so BΓ = Sm−1 × Sm−1 × · · · × Sm−1 (2k factors).

• In the proof of an analogue of Proposition 5.5, we replace a trivial family of (m+1)-handles

with that of m-handles.

Under this replacement, the proof of Theorem 2.3 works under the assumption of Theorem 6.1

completely without change.

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28 BORIS BOTVINNIK AND TADAYUKI WATANABE

7. Proof of Theorems 1.7 and 1.9

7.1. Recollection: admissible Morse functions. First we recall necessary definitions and re-

sults concerning Morse functions. To make all constructions more transparent, we restrict our

attention to the following special case.

Let W : X0 X1 be a (d + 1)-cobordism between two manifolds X0 and X1 with non-

empty boundaries. We assume, however, that the manifolds X0 and X1 have common boundary

∂X0 = Z = ∂1X1 , and the boundary ∂W is decomposed as

∂W = X0 ∪∂X0=Z×0 (Z × I) ∪∂X1=Z×1 X1.

We fix a reference Riemannian metric m on W and a small collar U ⊂ W of ∂W , such that it is

parametrized by (x, s) near X0 tX1 and by (z, s, t) near the cylinder Z × I . Here x ∈ X0 tX1 ,

where z ∈ Z and s ∈ I . Here we also identify ∂X0 = Z as x 7→ (z, 0, 0) and ∂X1 = Z as

x 7→ (z, 0, 1). Then we fix a linear function ξ0 : U → I given by (x, s) 7→ s near X0 t X1 and

(z, s, t) 7→ s near Z × I .

W Z × I

X0

X1

s

Figure 20. Cobordism W : X0 X1

By a Morse function on W we mean a Morse function f : W → [0, 1] such that

f−1(0) = ∂0W, f−1(1) = ∂1W,

and the restriction of f to the collar U coincides with the linear function ξ0 on U . We denote by

Cr(f) the set of critical points of f . By definition, Cr(f) ⊂W \ U .

We say that a Morse function f : W → [0, 1] is admissible if all its critical points have indices

at most (d − 2) (where dimW = d + 1). We denote by Morse(M) and Morseadm(W ) the spaces

of Morse functions and admissible Morse functions, respectively, which are equiped with the C∞ -

topologies.

Definition 7.1. Let f ∈ Morseadm(W ). A Riemannian metric m on W is compatible with the

Morse function f if for every critical point p ∈ Cr(f) with ind p = λ the positive and negative

eigenspaces TpW+ and TpW

− of the Hessian d2f are m-orthogonal, and d2f |TpW+ = m|TpW+ ,

d2f |TpW− = −m|TpW− .

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FAMILIES OF DIFFEOMORPHISMS AND CONCORDANCES 29

We notice that for a given Morse function f , the space of compatible metrics is convex. Thus

the space of pairs (f,m), where f ∈ Morseadm(W ), and m is a metric compatible with f , is

homotopy equivalent to the space Morseadm(W ). We call a pair (f,m) as above an admissible

Morse pair.

Theorem 7.2. [BHSW, Theorem 2.2] Let W : X0 X1 be a smooth compact cobordism with

∂W = X0 ∪ (Z × I) ∪X1

as above. Assume that g0 is a psc-metric on X0 and (f,m) is an admissible Morse pair on W .

Then there is a psc-metric g = g(g0, f,m) on W which extends g0 and has a product structure

near the boundary. In particular, g1 := g|X1 is a psc-metric.

7.2. Recollection: surgery for families of Morse functions. We recall a general setting from

[BHSW, Section 2.2]. A construction relevant to our goals leads to families of Morse functions, or

maps with fold singularities. Recall first a local description.

Definition 7.3. A map F : Rk ×Rd+1 → Rk ×R is called a standard map with a fold singularitiy

of index λ , if there is a c ∈ R so that f is given as

(4)Rk × Rd+1 −→ Rk × R,

(y, x) 7−→(y, c− x2

1 − · · · − x2λ + x2

λ+1 + · · ·+ x2d+1

).

Roughly speaking, the composition

Rk × Rd+1 F→ Rk × R p2→ R

with the projection p2 onto the second factor defines a Rk -parametrized family of Morse functions

of index λ on Rd+1 in standard form.

Let W : X0 X1 , dimW = d+ 1, be a cobordism between two manifolds with boundary as

above, i.e.,

(5) ∂W = X0 ∪∂X0=Z×0 (Z × I) ∪∂X1=Z×1 X1.

We denote by Difft(W ) the group of diffeomorphisms of W which restrict to the identity near

X0 ∪Z × I . There is a natural imbedding Diff∂(W ) ⊂ Difft(W ) which gives the fiber sequence of

corresponding classifying spaces:

BDiff∂(W )→ BDifft(W )→ BDiff∂(X1).

We consider a smooth fiber bundle π : E → B with fiber W , where B is a compact smooth

manifold, dimB = k and dimE = d + 1 + k . We assume that the structure group of this bundle

is the group Difft(W ).

Let π0 : E0 → B , π∂ : E∂ → B and π1 : E1 → B , be the restriction of the fiber bundle

π : E → B to the fibers X0 , Z × I and X1 respectively. Since each element of the structure

group Difft(W ) restricts to the identity near X0 and Z × I , the fiber bundle π0 : E0 → B and

π∂ : E∂ → B are trivialized:

E0 = B × ∂0Wπ0−→ B, E∂ = B × Z × I π∂−→ B.

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30 BORIS BOTVINNIK AND TADAYUKI WATANABE

We choose a splitting of the tangent bundle TE of the total space as TE ∼= π∗TB ⊕ Vert , where

Vert→ E is the bundle tangent to the fibers W , i.e. we choose a connection on TE .

Definition 7.4. Let π : E → B be a smooth bundle as above. For each b in B let

ib : Wb → E

be the inclusion of the fiber Wb := π−1(b). Let F : E → B × I be a smooth map. The map F is

said to be an admissible family of Morse functions or admissible with fold singularities with respect

to π if it satisfies the following conditions:

(1) The diagram

E B × I

B?

π

-F

p1

commutes. Here p1 : B × I → B is projection on the first factor.

(2) The pre-images F−1(B × 0) and F−1(B × 1) coincide with the submanifolds E0 and

E1 respectively.

(3) The set Cr(F ) ⊂ E of critical points of F is contained in E \ (E0 ∪ E∂ ∪ E1) and near

each critical point of F the bundle π is equivalent to the trivial bundle Rk × Rd+1 p1→ Rk

so that with respect to these coordinates on E and on B the map F is a standard map

Rk × Rd+1 → Rk × R with a fold singularity as in Definition 7.3

(4) For each z ∈ B the restriction

fb = F |Wb: Wb → b × I

p2−→ I

is an admissible Morse function, i.e. its critical points have indices ≤ d− 2.

Furthemore, we assume in addition that the smooth bundle π : E → B is a Riemannian

submersion π : (E,mE)→ (B,mB), see [Besse]. Here we denote by mE and mB the metrics on E

and B corresponding to the submersion π . Now let F : E → B×I be an admissible map with fold

singularities with respect to π as in Definition 7.4. If the restriction mb of the submersion metric

mE to each fiber Wb , b ∈ B , is compatible with the Morse function fb = F |Wb, we say that the

metric mE is compatible with the map F . Here is a relevant technical result:

Proposition 7.5. [BHSW, Proposition 2.8] Let π : E → B be a smooth bundle as above and

F : E → B × I be an admissible map with fold singularities with respect to π . Then the bundle

π : E → B admits the structure of a Riemannian submersion π : (E,mE)→ (B,mB) such that the

metric mE is compatible with the map F : E → B × I .

Let Cr(F ) be a the union of the critical loci of the function F . By definition, it splits into

finite number of folds Σ ⊂ Cr(F ). It is worth to recall that since the metric mE is a submersion

metric, the structure group of the vector bundle Vert → E is reduced to O(d + 1). Futhermore,

since the metrics mb are compatible with the Morse functions fb = F |Wb, the restriction Vert|Σ

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FAMILIES OF DIFFEOMORPHISMS AND CONCORDANCES 31

to a fold Σ ⊂ Cr(F ) splits further orthogonally into the positive and negative eigenspaces of the

Hessian of F . Thus the metric mE induces the splitting of the vector bundle

Vert|Σ ∼= Vert−Σ ⊕ Vert+Σ

with structure group O(p+ 1)×O(q+ 1) for each fold Σ, where p+ q+ 1 = d . This decomposition

plays a crucial role in the proof of the following result we need here:

Theorem 7.6. (cf. [BHSW, Theorem 2.9]) Let π : E → B be a smooth bundle with a (d + 1)-

cobordism W : X0 X1 between compact manifolds X0 and X1 with ∂X0 = Z = ∂X1 , and

∂W = X0 ∪∂X0=Z×0 (Z × I) ∪∂X1=Z×1 X1.

We assume that the structure group of the bundle π : E → B is Difft(W ) and the base space B is

a compact smooth simply connected manifold.

Let F : E → B× I be an admissible map with fold singularities with respect to π . In addition,

we assume that the fiber bundle π : E → B is given the structure of a Riemannian submersion

π : (E,mE) → (B,mB) such that the metric mE is compatible with the map F : E → B × I .

Finally, we assume that we are given a smooth map g0 : B → Rpsc(X0) with h0 = g0|Z .

Then there exists a Riemannian metric g = g(g0, F,mE) on E such that for each b ∈ B the

restriction g(b) = g|Wbto the fiber Wz = π−1(b) satisfies the following properies:

(1) g(b) extends g0(b);

(2) g(b) is a product metric gν(b) + ds2 near X0 tX1 , ν = 0, 1;

(3) g(b) is a product metric h0 + ds2 + dt2 near Z × I ;

(4) g(b) has positive scalar curvature on Wb .

Remark 7.7. The original theorem [BHSW, Theorem 2.9] assumes that the structure group is

Diff∂(W ); however, it is easy to see that the same proof works in more general situation, in

particular when the structure group is Difft(W ).

7.3. Back to the proof. In fact, it is enough to prove Theorem 1.9 which implies Theorem 1.7

as a simple corollary. Here we would like to work with both cases when d even and odd.

Now we consider bundles of h-cobordisms we have constructed. In both cases, when d is even

or odd, we obtain that such a bundle satisfies the conditions of Theorem 7.6. Thus we obtain that

every fiber has a psc-metric. This proves Theorems 1.7 and 1.9.

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Department of Mathematics, University of Oregon, Eugene, OR, 97405, USA

Email address: [email protected]

Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan

Email address: [email protected]