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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
FAMILIES OF DIFFEOMORPHISMS AND CONCORDANCES 3
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.
4 BORIS BOTVINNIK AND TADAYUKI WATANABE
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
-ι
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.
FAMILIES OF DIFFEOMORPHISMS AND CONCORDANCES 5
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
-ι
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
6 BORIS BOTVINNIK AND TADAYUKI WATANABE
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) .
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.
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
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.
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.
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 .
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.
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.
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.
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]).
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.
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 .
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
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.
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 .
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).
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),
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).
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))
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) .
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
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.
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− .
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.
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
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]