Surgery equivalence relations on 3-manifolds
Gwénaël Massuyeau(IMB, Dijon)

FJ-LMI Seminar, Tokyo
24 November 2023

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3-manifolds : examples
By “3-manifold”, we mean a topological 3-dimensional manifold, which iscompact, oriented and connected.

2/12

3-manifolds : examples
By “3-manifold”, we mean a topological 3-dimensional manifold, which iscompact, oriented and connected.
We consider 3-manifolds up to homeomorphism.

2/12

3-manifolds : examples
By “3-manifold”, we mean a topological 3-dimensional manifold, which iscompact, oriented and connected.
We consider 3-manifolds up to homeomorphism.

Example3-manifolds may have boundary :
1
,
D3

,
S1 × D2

···

g

···
handlebody of genus g

2/12

3-manifolds : examples
By “3-manifold”, we mean a topological 3-dimensional manifold, which iscompact, oriented and connected.
We consider 3-manifolds up to homeomorphism.

Example3-manifolds may have boundary :1,
D3| {z }
H0

,

{z
H1

g

···

S1 × D2|···

handlebody of genus g

}

|

{z
Hg

}

2/12

3-manifolds : examples
By “3-manifold”, we mean a topological 3-dimensional manifold, which iscompact, oriented and connected.
We consider 3-manifolds up to homeomorphism.

Example3-manifolds may have boundary :
···

1
,
D3

| {z }
H0,···

S1 × D2

|

{z
H1

g

handlebody of genus g

}

|

{z
Hg

}

. . . or they may be closed :S 3 = R3 ∪ {∞},RP 3 = S 3 /antipode,

S1 × S1 × S1
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3-manifolds : surgery operations
Given a 3-manifold M, we can construct new ones by surgery :3/12

3-manifolds : surgery operations
Given a 3-manifold M, we can construct new ones by surgery :choose a handlebody H ⊂ int(M);specify a homeomorphism h : ∂H → ∂H;

let MH := M \ int(H) ∪h H.
MH

M
h

H

H

3/12

3-manifolds : surgery operations
Given a 3-manifold M, we can construct new ones by surgery :choose a handlebody H ⊂ int(M);specify a homeomorphism h : ∂H → ∂H;

let MH := M \ int(H) ∪h H.
MH

M
h

H

H

ExampleWhen H has genus one, its “core” is a knot K in M, and the operation(MMH ) is called a Dehn surgery along K :3/12

3-manifolds : surgery operations
Given a 3-manifold M, we can construct new ones by surgery :choose a handlebody H ⊂ int(M);specify a homeomorphism h : ∂H → ∂H;

let MH := M \ int(H) ∪h H.
MH

M
h

H

H

ExampleWhen H has genus one, its “core” is a knot K in M, and the operation(MMH ) is called a Dehn surgery along K :K

H ⊂ S3(S 3 )H ∼=P(... for an appropriate h)

3/12

3-manifolds : surgery operations
Given a 3-manifold M, we can construct new ones by surgery :choose a handlebody H ⊂ int(M);specify a homeomorphism h : ∂H → ∂H;

let MH := M \ int(H) ∪h H.
MH

M
h

H

H

ExampleWhen H has genus one, its “core” is a knot K in M, and the operation(MMH ) is called a Dehn surgery along K :K

H ⊂ S3(S 3 )H ∼=P(... for an appropriate h)where P is the Poincaré homology 3-sphere : H∗ (P; Z) ' H∗ (S 3 ; Z).
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A classical result in low-dimensional topology
Theorem (Heegaard 1898)
Up to homeomorphism, any two 3-manifolds M and N (with the “same”
boundary) can be obtained one from the other by a single surgery.

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A classical result in low-dimensional topology
Theorem (Heegaard 1898)
Up to homeomorphism, any two 3-manifolds M and N (with the “same”
boundary) can be obtained one from the other by a single surgery.

N.B. Besides, M and N are related one to the other by finitely many
Dehn surgeries (Rochlin 1951, Thom 1951, Wallace 1960, Lickorish 1964).

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A classical result in low-dimensional topology
Theorem (Heegaard 1898)
Up to homeomorphism, any two 3-manifolds M and N (with the “same”
boundary) can be obtained one from the other by a single surgery.

N.B. Besides, M and N are related one to the other by finitely many
Dehn surgeries (Rochlin 1951, Thom 1951, Wallace 1960, Lickorish 1964).
Assume that ∂M = ∂N = ∅.

4/12

A classical result in low-dimensional topology
Theorem (Heegaard 1898)
Up to homeomorphism, any two 3-manifolds M and N (with the “same”
boundary) can be obtained one from the other by a single surgery.

N.B. Besides, M and N are related one to the other by finitely many
Dehn surgeries (Rochlin 1951, Thom 1951, Wallace 1960, Lickorish 1964).
Assume that ∂M = ∂N = ∅.
Fact. M has a Heegaard splitting, i.e. there are handlebodies H, H 0 ⊂ M(of the same genus) such that M = H ∪ H 0 and H ∩ H 0 = ∂H = ∂H 0 .

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A classical result in low-dimensional topology
Theorem (Heegaard 1898)
Up to homeomorphism, any two 3-manifolds M and N (with the “same”
boundary) can be obtained one from the other by a single surgery.

N.B. Besides, M and N are related one to the other by finitely many
Dehn surgeries (Rochlin 1951, Thom 1951, Wallace 1960, Lickorish 1964).
Assume that ∂M = ∂N = ∅.
Fact. M has a Heegaard splitting, i.e. there are handlebodies H, H 0 ⊂ M(of the same genus) such that M = H ∪ H 0 and H ∩ H 0 = ∂H = ∂H 0 .
Indeed, M has a triangulation (Moise 1952) :4/12

A classical result in low-dimensional topology
Theorem (Heegaard 1898)
Up to homeomorphism, any two 3-manifolds M and N (with the “same”
boundary) can be obtained one from the other by a single surgery.

N.B. Besides, M and N are related one to the other by finitely many
Dehn surgeries (Rochlin 1951, Thom 1951, Wallace 1960, Lickorish 1964).
Assume that ∂M = ∂N = ∅.
Fact. M has a Heegaard splitting, i.e. there are handlebodies H, H 0 ⊂ M(of the same genus) such that M = H ∪ H 0 and H ∩ H 0 = ∂H = ∂H 0 .
Indeed, M has a triangulation (Moise 1952) :4/12

A classical result in low-dimensional topology
Theorem (Heegaard 1898)
Up to homeomorphism, any two 3-manifolds M and N (with the “same”
boundary) can be obtained one from the other by a single surgery.

N.B. Besides, M and N are related one to the other by finitely many
Dehn surgeries (Rochlin 1951, Thom 1951, Wallace 1960, Lickorish 1964).
Assume that ∂M = ∂N = ∅.
Fact. M has a Heegaard splitting, i.e. there are handlebodies H, H 0 ⊂ M(of the same genus) such that M = H ∪ H 0 and H ∩ H 0 = ∂H = ∂H 0 .
Indeed, M has a triangulation (Moise 1952) :Take H :=(neighborhood of vertices & edges) , H 0 := M \ int(H).

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A classical result in low-dimensional topology: proof
Theorem (Heegaard 1898)
Up to homeomorphism, any two 3-manifolds M and N (with the “same”
boundary) can be obtained one from the other by a single surgery.
Proof.

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A classical result in low-dimensional topology: proof
Theorem (Heegaard 1898)
Up to homeomorphism, any two 3-manifolds M and N (with the “same”
boundary) can be obtained one from the other by a single surgery.
Proof. Consider some Heegaard splittings :M = H ∪ H 0 of genus g ,N = J ∪ J 0 of genus k.

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A classical result in low-dimensional topology: proof
Theorem (Heegaard 1898)
Up to homeomorphism, any two 3-manifolds M and N (with the “same”
boundary) can be obtained one from the other by a single surgery.
Proof. Consider some Heegaard splittings :M = H ∪ H 0 of genus g ,N = J ∪ J 0 of genus k.

We can assume that g = k by “stabilizing” the splitting of smaller genus :
H0

H̃ 0
here

···
H

···
H̃

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A classical result in low-dimensional topology: proof
Theorem (Heegaard 1898)
Up to homeomorphism, any two 3-manifolds M and N (with the “same”
boundary) can be obtained one from the other by a single surgery.
Proof. Consider some Heegaard splittings :M = H ∪ H 0 of genus g ,N = J ∪ J 0 of genus k.

We can assume that g = k by “stabilizing” the splitting of smaller genus :
H0

H̃ 0
here

···
H

···
H̃

Fix some homeomorphisms ψ : H → J and ψ 0 : H 0 → J 0 :5/12

A classical result in low-dimensional topology: proof
Theorem (Heegaard 1898)
Up to homeomorphism, any two 3-manifolds M and N (with the “same”
boundary) can be obtained one from the other by a single surgery.
Proof. Consider some Heegaard splittings :M = H ∪ H 0 of genus g ,N = J ∪ J 0 of genus k.

We can assume that g = k by “stabilizing” the splitting of smaller genus :
H0

H̃ 0
here

···
H

···
H̃

Fix some homeomorphisms ψ : H → J and ψ 0 : H 0 → J 0 :
∂H

ψ
∼
=

/ ∂J

∂J 0

(ψ 0 )−1/ ∂H 0
∼
=

7 ∂H

h
5/12

A classical result in low-dimensional topology: proof
Theorem (Heegaard 1898)
Up to homeomorphism, any two 3-manifolds M and N (with the “same”
boundary) can be obtained one from the other by a single surgery.
Proof. Consider some Heegaard splittings :M = H ∪ H 0 of genus g ,N = J ∪ J 0 of genus k.

We can assume that g = k by “stabilizing” the splitting of smaller genus :
H0

H̃ 0
here

···
H

···
H̃

Fix some homeomorphisms ψ : H → J and ψ 0 : H 0 → J 0 :
ψ
∼
=

∂H

/ ∂J

∂J 0

(ψ 0 )−1/ ∂H 0
∼
=

7 ∂H

h
0

Assemble ψ and ψ to a homeomorphism Ψ : MH → N.



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Defining non-trivial surgery equivalence relations

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Defining non-trivial surgery equivalence relations

Thus, surgery (MMH ) is a very permissive way to modify 3-manifolds.

6/12

Defining non-trivial surgery equivalence relations

Thus, surgery (MMH ) is a very permissive way to modify 3-manifolds.

To define non-trivial equivalence relations on the set of (homeomorphismclasses of) 3-manifolds,

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Defining non-trivial surgery equivalence relations

Thus, surgery (MMH ) is a very permissive way to modify 3-manifolds.

To define non-trivial equivalence relations on the set of (homeomorphismclasses of) 3-manifolds, one can either(i) restrict the type of the handlebody H in M,

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Defining non-trivial surgery equivalence relations

Thus, surgery (MMH ) is a very permissive way to modify 3-manifolds.

To define non-trivial equivalence relations on the set of (homeomorphismclasses of) 3-manifolds, one can either(i) restrict the type of the handlebody H in M,(ii) or, restrict the type of the gluing homeomorphism h : ∂H → ∂H.

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Defining non-trivial surgery equivalence relations

Thus, surgery (MMH ) is a very permissive way to modify 3-manifolds.

To define non-trivial equivalence relations on the set of (homeomorphismclasses of) 3-manifolds, one can either(i) restrict the type of the handlebody H in M,(ii) or, restrict the type of the gluing homeomorphism h : ∂H → ∂H.
ExampleIn option (i) with Dehn surgeries, one may impose conditions on thehomotopy classes of knots (Cochran–Gerges–Orr 2001).

6/12

Defining non-trivial surgery equivalence relations

Thus, surgery (MMH ) is a very permissive way to modify 3-manifolds.

To define non-trivial equivalence relations on the set of (homeomorphismclasses of) 3-manifolds, one can either(i) restrict the type of the handlebody H in M,(ii) or, restrict the type of the gluing homeomorphism h : ∂H → ∂H.
ExampleIn option (i) with Dehn surgeries, one may impose conditions on thehomotopy classes of knots (Cochran–Gerges–Orr 2001).

We choose option (ii) and observe the following : up to homeomorphism,
MH only depends on the isotopy class of h.

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The mapping class group of a surface
Σ : compact, oriented surface with one boundary component

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The mapping class group of a surface
Σ : compact, oriented surface with one boundary component
The mapping class group of Σ isM(Σ) :={ homeomorphisms Σ → Σ that fix ∂Σ pointwise }isotopy

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The mapping class group of a surface
Σ : compact, oriented surface with one boundary component
The mapping class group of Σ isM(Σ) :={ homeomorphisms Σ → Σ that fix ∂Σ pointwise }isotopyand the Torelli group of Σ is I(Σ) := ker M(Σ) → Aut(H1 (Σ; Z) .

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The mapping class group of a surface
Σ : compact, oriented surface with one boundary component
The mapping class group of Σ isM(Σ) :={ homeomorphisms Σ → Σ that fix ∂Σ pointwise }isotopyand the Torelli group of Σ is I(Σ) := ker M(Σ) → Aut(H1 (Σ; Z) .
M(Σ) acts on π := π1 (Σ),

7/12The mapping class group of a surface
Σ : compact, oriented surface with one boundary component
The mapping class group of Σ isM(Σ) :={ homeomorphisms Σ → Σ that fix ∂Σ pointwise }isotopyand the Torelli group of Σ is I(Σ) := ker M(Σ) → Aut(H1 (Σ; Z) .
M(Σ) acts on π := π1 (Σ), and its lower central series :π = Γ1 π ⊃ Γ2 π ⊃ · · · ⊃ Γk π ⊃ · · ·where Γk π := [Γk−1 π, π]

7/12The mapping class group of a surface
Σ : compact, oriented surface with one boundary component
The mapping class group of Σ isM(Σ) :={ homeomorphisms Σ → Σ that fix ∂Σ pointwise }isotopyand the Torelli group of Σ is I(Σ) := ker M(Σ) → Aut(H1 (Σ; Z) .
M(Σ) acts on π := π1 (Σ), and its lower central series :π = Γ1 π ⊃ Γ2 π ⊃ · · · ⊃ Γk π ⊃ · · ·where Γk π := [Γk−1 π, π]

We are interested in the Johnson filtration of M := M(Σ) :M = J0 M ⊃ J1 M ⊃ · · · ⊃ Jk M ⊃ · · ·| {z }
I
where Jk M := ker M → Aut(π/Γk+1 π)

7/12The mapping class group of a surface
Σ : compact, oriented surface with one boundary component
The mapping class group of Σ isM(Σ) :={ homeomorphisms Σ → Σ that fix ∂Σ pointwise }isotopyand the Torelli group of Σ is I(Σ) := ker M(Σ) → Aut(H1 (Σ; Z) .
M(Σ) acts on π := π1 (Σ), and its lower central series :π = Γ1 π ⊃ Γ2 π ⊃ · · · ⊃ Γk π ⊃ · · ·where Γk π := [Γk−1 π, π]

We are interested in the Johnson filtration of M := M(Σ) :M = J0 M ⊃ J1 M ⊃ · · · ⊃ Jk M ⊃ · · ·| {z }
I
where Jk M := ker M → Aut(π/Γk+1 π)
Properties. (Morita 1993)
Tk≥1 Jk M = {1}.
[Jk M, J` M] ⊂ Jk+` Mhence Γm I ⊂ Jm M.

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Jk -equivalence and Yk -equivalence
Definition
M, N : 3-manifolds (with the “same” boundary)

8/12

Jk -equivalence and Yk -equivalence
Definition
M, N : 3-manifolds (with the “same” boundary)
M is Jk -equivalent (resp. Yk -equivalent) to N if there is an H ⊂ int(M)and an h ∈ Jk M(∂H) (resp. an h ∈ Γk I(∂H)) such that M
MH ∼
= N.

8/12

Jk -equivalence and Yk -equivalence
Definition
M, N : 3-manifolds (with the “same” boundary)
M is Jk -equivalent (resp. Yk -equivalent) to N if there is an H ⊂ int(M)and an h ∈ Jk M(∂H) (resp. an h ∈ Γk I(∂H)) such that M
MH ∼
= N.
Y1 -eq. ⇐= Y2 -eq. ⇐= Y3 -eq. ⇐= · · ·k⇓⇓
J1 -eq. ⇐= J2 -eq. ⇐= J3 -eq. ⇐= · · ·Yk -eq. ⇐= · · ·⇓Jk -eq. ⇐= · · ·8/12

Jk -equivalence and Yk -equivalence
Definition
M, N : 3-manifolds (with the “same” boundary)
M is Jk -equivalent (resp. Yk -equivalent) to N if there is an H ⊂ int(M)and an h ∈ Jk M(∂H) (resp. an h ∈ Γk I(∂H)) such that M
MH ∼
= N.
Y1 -eq. ⇐= Y2 -eq. ⇐= Y3 -eq. ⇐= · · ·k⇓⇓
J1 -eq. ⇐= J2 -eq. ⇐= J3 -eq. ⇐= · · ·Yk -eq. ⇐= · · ·⇓Jk -eq. ⇐= · · ·These equivalence relations preserve the homology type of 3-manifolds.

8/12

Jk -equivalence and Yk -equivalence
Definition
M, N : 3-manifolds (with the “same” boundary)
M is Jk -equivalent (resp. Yk -equivalent) to N if there is an H ⊂ int(M)and an h ∈ Jk M(∂H) (resp. an h ∈ Γk I(∂H)) such that M
MH ∼
= N.
Y1 -eq. ⇐= Y2 -eq. ⇐= Y3 -eq. ⇐= · · ·k⇓⇓
J1 -eq. ⇐= J2 -eq. ⇐= J3 -eq. ⇐= · · ·Yk -eq. ⇐= · · ·⇓Jk -eq. ⇐= · · ·These equivalence relations preserve the homology type of 3-manifolds.
ProblemClassify/characterize them with topological invariants of 3-manifolds !

8/12

Jk -equivalence and Yk -equivalence
Definition
M, N : 3-manifolds (with the “same” boundary)
M is Jk -equivalent (resp. Yk -equivalent) to N if there is an H ⊂ int(M)and an h ∈ Jk M(∂H) (resp. an h ∈ Γk I(∂H)) such that M
MH ∼
= N.
Y1 -eq. ⇐= Y2 -eq. ⇐= Y3 -eq. ⇐= · · ·k⇓⇓
J1 -eq. ⇐= J2 -eq. ⇐= J3 -eq. ⇐= · · ·Yk -eq. ⇐= · · ·⇓Jk -eq. ⇐= · · ·These equivalence relations preserve the homology type of 3-manifolds.
ProblemClassify/characterize them with topological invariants of 3-manifolds !
The Yk -eq. relations are more “manageable” than the Jk -eq. relations :8/12

Jk -equivalence and Yk -equivalence
Definition
M, N : 3-manifolds (with the “same” boundary)
M is Jk -equivalent (resp. Yk -equivalent) to N if there is an H ⊂ int(M)and an h ∈ Jk M(∂H) (resp. an h ∈ Γk I(∂H)) such that M
MH ∼
= N.
Y1 -eq. ⇐= Y2 -eq. ⇐= Y3 -eq. ⇐= · · ·k⇓⇓
J1 -eq. ⇐= J2 -eq. ⇐= J3 -eq. ⇐= · · ·Yk -eq. ⇐= · · ·⇓Jk -eq. ⇐= · · ·These equivalence relations preserve the homology type of 3-manifolds.
ProblemClassify/characterize them with topological invariants of 3-manifolds !
The Yk -eq. relations are more “manageable” than the Jk -eq. relations :1
the “clasper calculus” (Goussarov, Habiro) applies to their study;
8/12

Jk -equivalence and Yk -equivalence
Definition
M, N : 3-manifolds (with the “same” boundary)
M is Jk -equivalent (resp. Yk -equivalent) to N if there is an H ⊂ int(M)and an h ∈ Jk M(∂H) (resp. an h ∈ Γk I(∂H)) such that M
MH ∼
= N.
Y1 -eq. ⇐= Y2 -eq. ⇐= Y3 -eq. ⇐= · · ·k⇓⇓
J1 -eq. ⇐= J2 -eq. ⇐= J3 -eq. ⇐= · · ·Yk -eq. ⇐= · · ·⇓Jk -eq. ⇐= · · ·These equivalence relations preserve the homology type of 3-manifolds.
ProblemClassify/characterize them with topological invariants of 3-manifolds !
The Yk -eq. relations are more “manageable” than the Jk -eq. relations :1
the “clasper calculus” (Goussarov, Habiro) applies to their study;
2
they have much to do with “finite-type invariants”.
8/12

Finite-type invariants of 3-manifolds
fReminder. (Fréchet 1909) A continuous function Rn −→ R is polynomialof degree ≤ d if, and only if, for any x0 , . . . , xd ∈ Rn , we haveX X(−1)|S| · f
xs = 0 ∈ R.
S⊂{0,...,d}

s∈S

9/12

Finite-type invariants of 3-manifolds
fReminder. (Fréchet 1909) A continuous function Rn −→ R is polynomialof degree ≤ d if, and only if, for any x0 , . . . , xd ∈ Rn , we haveX X(−1)|S| · f
xs = 0 ∈ R.
S⊂{0,...,d}s∈S

Definition (Ohtsuki, Garoufalidis–Levine, Goussarov, Habiro)
F : an invariant of 3-manifolds with values in an abelian group A

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Finite-type invariants of 3-manifolds
fReminder. (Fréchet 1909) A continuous function Rn −→ R is polynomialof degree ≤ d if, and only if, for any x0 , . . . , xd ∈ Rn , we haveX X(−1)|S| · f
xs = 0 ∈ R.
S⊂{0,...,d}s∈S

Definition (Ohtsuki, Garoufalidis–Levine, Goussarov, Habiro)
F : an invariant of 3-manifolds with values in an abelian group A
F is a finite-type invariant of degree ≤ d if, for any M, any disjoint
H0 , . . . , Hd ⊂ int(M) and any h0 ∈ I(∂H0 ), . . . , hd ∈ I(∂Hd ), we haveX(−1)|S| · F M Ss∈S Hs = 0 ∈ A.
S⊂{0,...,d}9/12

Finite-type invariants of 3-manifolds
fReminder. (Fréchet 1909) A continuous function Rn −→ R is polynomialof degree ≤ d if, and only if, for any x0 , . . . , xd ∈ Rn , we haveX X(−1)|S| · f
xs = 0 ∈ R.
S⊂{0,...,d}s∈S

Definition (Ohtsuki, Garoufalidis–Levine, Goussarov, Habiro)
F : an invariant of 3-manifolds with values in an abelian group A
F is a finite-type invariant of degree ≤ d if, for any M, any disjoint
H0 , . . . , Hd ⊂ int(M) and any h0 ∈ I(∂H0 ), . . . , hd ∈ I(∂Hd ), we haveX(−1)|S| · F M Ss∈S Hs = 0 ∈ A.
S⊂{0,...,d}Fact. For any 3 manifolds M and N (with “same” boundary), we haveM ∼Yd+1 N =⇒ F (M) = F (N) for all finite-type inv. F of deg. ≤ d
9/12

Finite-type invariants of 3-manifolds
fReminder. (Fréchet 1909) A continuous function Rn −→ R is polynomialof degree ≤ d if, and only if, for any x0 , . . . , xd ∈ Rn , we haveX X(−1)|S| · f
xs = 0 ∈ R.
S⊂{0,...,d}s∈S

Definition (Ohtsuki, Garoufalidis–Levine, Goussarov, Habiro)
F : an invariant of 3-manifolds with values in an abelian group A
F is a finite-type invariant of degree ≤ d if, for any M, any disjoint
H0 , . . . , Hd ⊂ int(M) and any h0 ∈ I(∂H0 ), . . . , hd ∈ I(∂Hd ), we haveX(−1)|S| · F M Ss∈S Hs = 0 ∈ A.
S⊂{0,...,d}Fact. For any 3 manifolds M and N (with “same” boundary), we haveM ∼Yd+1 N =⇒ F (M) = F (N) for all finite-type inv. F of deg. ≤ d
Hopefully, quantum topology provides a plethora of finite-type invariants !
9/12

The case of homology cylinders : the LMO homomorphism
Σ: compact, oriented surface with one boundary component

10/12

The case of homology cylinders : the LMO homomorphism
Σ: compact, oriented surface with one boundary componentIC(Σ) := homology cylinders over Σ ∼
=

10/12

The case of homology cylinders : the LMO homomorphism
Σ: compact, oriented surface with one boundary componentIC(Σ) := homology cylinders over Σ ∼=- same boundary and same homology type as Σ × [0, 1]

10/12

The case of homology cylinders : the LMO homomorphism
Σ: compact, oriented surface with one boundary componentIC(Σ) := homology cylinders over Σ ∼=- same boundary and same homology type as Σ × [0, 1]
IC(Σ) is a monoid, such that IC(Σ)× ' I(Σ).

10/12The case of homology cylinders : the LMO homomorphism
Σ: compact, oriented surface with one boundary componentIC(Σ) := homology cylinders over Σ ∼=- same boundary and same homology type as Σ × [0, 1]
IC(Σ) is a monoid, such that IC(Σ)× ' I(Σ).
ExampleIC(D 2 ) is essentially the monoid of homology 3-spheres.

10/12

The case of homology cylinders : the LMO homomorphism
Σ: compact, oriented surface with one boundary componentIC(Σ) := homology cylinders over Σ ∼=- same boundary and same homology type as Σ × [0, 1]
IC(Σ) is a monoid, such that IC(Σ)× ' I(Σ).
ExampleIC(D 2 ) is essentially the monoid of homology 3-spheres.
Theorem (Cheptea–Habiro–M 2008, Habiro–M 2009)
The Le–Murakami–Ohtsuki invariant of homology 3-spheres extends to ahomomorphism
Z : IC(Σ) −→ A H1 (Σ; Q)where A H1 (Σ; Q) := trivalent graphs with “legs” colored by H1 (Σ; Q)
is equipped with a diagrammatic analogue of the Moyal–Weyl product.
c

d

g

e

f

h

k

i

j

b
a

a, b, . . . , k ∈ H1 (Σ; Q)

10/12

The case of homology cylinders : the LMO homomorphism
Σ: compact, oriented surface with one boundary componentIC(Σ) := homology cylinders over Σ ∼=- same boundary and same homology type as Σ × [0, 1]
IC(Σ) is a monoid, such that IC(Σ)× ' I(Σ).
ExampleIC(D 2 ) is essentially the monoid of homology 3-spheres.
Theorem (Cheptea–Habiro–M 2008, Habiro–M 2009)
The Le–Murakami–Ohtsuki invariant of homology 3-spheres extends to ahomomorphism
Z : IC(Σ) −→ A H1 (Σ; Q)where A H1 (Σ; Q) := trivalent graphs with “legs” colored by H1 (Σ; Q)
is equipped with a diagrammatic analogue of the Moyal–Weyl product.
c

d

g

e

f

h

k

i

j

b
a

a, b, . . . , k ∈ H1 (Σ; Q)

Furthermore, Z is universal among Q-valued finite-type invariants.
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The case of homology cylinders : some results
The LMO homomorphism Z is a powerful tool to classify relations :11/12

The case of homology cylinders : some results
The LMO homomorphism Z is a powerful tool to classify relations :the Yk -eq. and Jk -eq. up to k = 3 (M–Meilhan 2013);
the Y4 -equivalence too (Nozaki–Sato–Suzuki 2022).

11/12

The case of homology cylinders : some results
The LMO homomorphism Z is a powerful tool to classify relations :the Yk -eq. and Jk -eq. up to k = 3 (M–Meilhan 2013);
the Y4 -equivalence too (Nozaki–Sato–Suzuki 2022).
But, there are still open questions in arbitrary high degrees :11/12

The case of homology cylinders : some results
The LMO homomorphism Z is a powerful tool to classify relations :the Yk -eq. and Jk -eq. up to k = 3 (M–Meilhan 2013);
the Y4 -equivalence too (Nozaki–Sato–Suzuki 2022).
But, there are still open questions in arbitrary high degrees :Goussarov–Habiro conjecture
Let d ∈ N. For any M, N ∈ IC(Σ), we haveM ∼Yd+1 N ⇐⇒ F (M) = F (N) for all finite-type inv. F of deg. ≤ d

11/12

The case of homology cylinders : some results
The LMO homomorphism Z is a powerful tool to classify relations :the Yk -eq. and Jk -eq. up to k = 3 (M–Meilhan 2013);
the Y4 -equivalence too (Nozaki–Sato–Suzuki 2022).
But, there are still open questions in arbitrary high degrees :Goussarov–Habiro conjecture
Let d ∈ N. For any M, N ∈ IC(Σ), we haveM ∼Yd+1 N ⇐⇒ F (M) = F (N) for all finite-type inv. F of deg. ≤ d
OK for : Σ = D 2 (Habiro 2000); in a weaker form (M 2007);
up to degree d = 4 (Nozaki–Sato–Suzuki 2022).

11/12

The case of homology cylinders : some results
The LMO homomorphism Z is a powerful tool to classify relations :the Yk -eq. and Jk -eq. up to k = 3 (M–Meilhan 2013);
the Y4 -equivalence too (Nozaki–Sato–Suzuki 2022).
But, there are still open questions in arbitrary high degrees :Goussarov–Habiro conjecture
Let d ∈ N. For any M, N ∈ IC(Σ), we haveM ∼Yd+1 N ⇐⇒ F (M) = F (N) for all finite-type inv. F of deg. ≤ d
OK for : Σ = D 2 (Habiro 2000); in a weaker form (M 2007);
up to degree d = 4 (Nozaki–Sato–Suzuki 2022).
Question
Up to which k ≥ 1, any homology 3-sphere is Jk -equivalent to S 3 ?

11/12

The case of homology cylinders : some results
The LMO homomorphism Z is a powerful tool to classify relations :the Yk -eq. and Jk -eq. up to k = 3 (M–Meilhan 2013);
the Y4 -equivalence too (Nozaki–Sato–Suzuki 2022).
But, there are still open questions in arbitrary high degrees :Goussarov–Habiro conjecture
Let d ∈ N. For any M, N ∈ IC(Σ), we haveM ∼Yd+1 N ⇐⇒ F (M) = F (N) for all finite-type inv. F of deg. ≤ d
OK for : Σ = D 2 (Habiro 2000); in a weaker form (M 2007);
up to degree d = 4 (Nozaki–Sato–Suzuki 2022).
Question
Up to which k ≥ 1, any homology 3-sphere is Jk -equivalent to S 3 ?
OK for : k = 2 (Morita 1989) ; k = 3 (Pitsch 2008) ; k = 4 (Faes 2022).
11/12

The case of homology cylinders : some results
The LMO homomorphism Z is a powerful tool to classify relations :the Yk -eq. and Jk -eq. up to k = 3 (M–Meilhan 2013);
the Y4 -equivalence too (Nozaki–Sato–Suzuki 2022).
But, there are still open questions in arbitrary high degrees :Goussarov–Habiro conjecture
Let d ∈ N. For any M, N ∈ IC(Σ), we haveM ∼Yd+1 N ⇐⇒ F (M) = F (N) for all finite-type inv. F of deg. ≤ d
OK for : Σ = D 2 (Habiro 2000); in a weaker form (M 2007);
up to degree d = 4 (Nozaki–Sato–Suzuki 2022).
Question
Up to which k ≥ 1, any homology 3-sphere is Jk -equivalent to S 3 ?
OK for : k = 2 (Morita 1989) ; k = 3 (Pitsch 2008) ; k = 4 (Faes 2022).
Not OK for k = 5 ? (Pitsch–Riba 2023)
11/12

For further information, here is a survey paper :arXiv:2301.12428

12/12

For further information, here is a survey paper :arXiv:2301.12428

Thanks for your attention !

12/12

For further information, here is a survey paper :arXiv:2301.12428

Thanks for your attention !
Merci de votre attention !

12/12

For further information, here is a survey paper :arXiv:2301.12428

Thanks for your attention !
Merci de votre attention !
ご清聴ありがとうございました

12/12