COBORDISMS OF LEFSCHETZ FIBRATIONS ON 4-MANIFOLDS...
Transcript of COBORDISMS OF LEFSCHETZ FIBRATIONS ON 4-MANIFOLDS...
Knots and Low Dimensional TopologyA Satellite Conference of Seoul ICM 2014
August 22-26, 2014
COBORDISMS OF LEFSCHETZ FIBRATIONS
ON 4-MANIFOLDS
Daniele ZuddasKorea Institute for Advanced Study
http://newton.kias.re.kr/~zuddas [email protected]
1(Generalized) Lefschetz fibrations
f : V m+2k ⇧Mm+2
s.t. at critical points is locally equivalent to the map
Rm ⇥ Ck ⇧ Rm ⇥ C(x, z1, . . . , zk) ⇧ (x, z2
1 + · · · + z2k)
Away from Crit(f) ⌅M , f is a fiber bundlecodimCrit(f) = 2.
m = 0 ⌃⌥ ordinary Lefschetz fibration.
We assume k = 2⌃⌥ the regular fiber is a surface Fg of some genus g.
The monodromy of a meridian of Crit(f) is a Dehn twist about a curve c ⌅ Fg.
We have the monodromy representation
⇤f : ⇥1(M � Crit(f))⇧Modg .
1(Generalized) Lefschetz fibrations
f : V m+2k ⇧Mm+2
s.t. at critical points is locally equivalent to the map
Rm ⇥ Ck ⇧ Rm ⇥ C(x, z1, . . . , zk) ⇧ (x, z2
1 + · · · + z2k)
Away from Crit(f) ⌅M , f is a fiber bundlecodimCrit(f) = 2.
m = 0 ⌃⌥ ordinary Lefschetz fibration.
We assume k = 2⌃⌥ the regular fiber is a surface Fg of some genus g.
The monodromy of a meridian of Crit(f) is a Dehn twist about a curve c ⌅ Fg.
We have the monodromy representation
⇤f : ⇥1(M � Crit(f))⇧Modg .
1(Generalized) Lefschetz fibrations
f : V m+2k ⇧Mm+2
s.t. at critical points is locally equivalent to the map
Rm ⇥ Ck ⇧ Rm ⇥ C(x, z1, . . . , zk) ⇧ (x, z2
1 + · · · + z2k)
Away from Crit(f) ⌅M , f is a fiber bundlecodimCrit(f) = 2.
m = 0 ⌃⌥ ordinary Lefschetz fibration.
We assume k = 2⌃⌥ the regular fiber is a surface Fg of some genus g.
The monodromy of a meridian of Crit(f) is a Dehn twist about a curve c ⌅ Fg.
We have the monodromy representation
⇤f : ⇥1(M � Crit(f))⇧Modg .
1(Generalized) Lefschetz fibrations
f : V m+2k ⇧Mm+2
s.t. at critical points is locally equivalent to the map
Rm ⇥ Ck ⇧ Rm ⇥ C(x, z1, . . . , zk) ⇧ (x, z2
1 + · · · + z2k)
Away from Crit(f) ⌅M , f is a fiber bundlecodimCrit(f) = 2.
m = 0 ⌃⌥ ordinary Lefschetz fibration.
We assume k = 2⌃⌥ the regular fiber is a surface Fg of some genus g.
The monodromy of a meridian of Crit(f) is a Dehn twist about a curve c ⌅ Fg.
We have the monodromy representation
⇤f : ⇥1(M � Crit(f))⇧Modg .
1(Generalized) Lefschetz fibrations
f : V m+2k ⇧Mm+2
s.t. at critical points is locally equivalent to the map
Rm ⇥ Ck ⇧ Rm ⇥ C(x, z1, . . . , zk) ⇧ (x, z2
1 + · · · + z2k)
Away from Crit(f) ⌅M , f is a fiber bundlecodimCrit(f) = 2.
m = 0 ⌃⌥ ordinary Lefschetz fibration.
We assume k = 2⌃⌥ the regular fiber is a surface Fg of some genus g.
The monodromy of a meridian of Crit(f) is a Dehn twist about a curve c ⌅ Fg.
We have the monodromy representation
⇤f : ⇥1(M � Crit(f))⇧Modg .
1(Generalized) Lefschetz fibrations
f : V m+2k ⇧Mm+2
s.t. at critical points is locally equivalent to the map
Rm ⇥ Ck ⇧ Rm ⇥ C(x, z1, . . . , zk) ⇧ (x, z2
1 + · · · + z2k)
Away from Crit(f) ⌅M , f is a fiber bundlecodimCrit(f) = 2.
m = 0 ⌃⌥ ordinary Lefschetz fibration.
We assume k = 2⌃⌥ the regular fiber is a surface Fg of some genus g.
The monodromy of a meridian of Crit(f) is a Dehn twist about a curve c ⌅ Fg.
We have the monodromy representation
⇤f : ⇥1(M � Crit(f))⇧Modg .
1(Generalized) Lefschetz fibrations
f : V m+2k ⇧Mm+2
s.t. at critical points is locally equivalent to the map
Rm ⇥ Ck ⇧ Rm ⇥ C(x, z1, . . . , zk) ⇧ (x, z2
1 + · · · + z2k)
Away from Crit(f) ⌅M , f is a fiber bundlecodimCrit(f) = 2.
m = 0 ⌃⌥ ordinary Lefschetz fibration.
We assume k = 2⌃⌥ the regular fiber is a surface Fg of some genus g.
The monodromy of a meridian of Crit(f) is a Dehn twist about a curve c ⌅ Fg.
We have the monodromy representation
⇤f : ⇥1(M � Crit(f))⇧Modg .
2Pullbacks
V
f
MN q
Vq
V
q∗(f)
Pullbackof f
f -regular
Fiberpreserving
iff q and q|∂N
transverse to f
q : N ⇧M is f -regular i⇤ q and q|�N are transverse to f .
�V = {(x, v) � N ⇥ V | q(x) = f(v)}
(q�(f))(x, v) = x
�q(x, v) = v
q�(f) has the same fiber of f .
2Pullbacks
V
f
MN q
Vq
V
q∗(f)
Pullbackof f
f -regular
Fiberpreserving
iff q and q|∂N
transverse to f
q : N ⇧M is f -regular i⇤ q and q|�N are transverse to f .
�V = {(x, v) � N ⇥ V | q(x) = f(v)}
(q�(f))(x, v) = x
�q(x, v) = v
q�(f) has the same fiber of f .
2Pullbacks
V
f
MN q
Vq
V
q∗(f)
Pullbackof f
f -regular
Fiberpreserving
iff q and q|∂N
transverse to f
q : N ⇧M is f -regular i⇤ q and q|�N are transverse to f .
�V = {(x, v) � N ⇥ V | q(x) = f(v)}
(q�(f))(x, v) = x
�q(x, v) = v
q�(f) has the same fiber of f .
2Pullbacks
V
f
MN q
Vq
V
q∗(f)
Pullbackof f
f -regular
Fiberpreserving
iff q and q|∂N
transverse to f
q : N ⇧M is f -regular i⇤ q and q|�N are transverse to f .
�V = {(x, v) � N ⇥ V | q(x) = f(v)}
(q�(f))(x, v) = x
�q(x, v) = v
q�(f) has the same fiber of f .
2Pullbacks
V
f
MN q
Vq
V
q∗(f)
Pullbackof f
f -regular
Fiberpreserving
iff q and q|∂N
transverse to f
q : N ⇧M is f -regular i⇤ q and q|�N are transverse to f .
�V = {(x, v) � N ⇥ V | q(x) = f(v)}
(q�(f))(x, v) = x
�q(x, v) = v
q�(f) has the same fiber of f .
2Pullbacks
V
f
MN q
Vq
V
q∗(f)
Pullbackof f
f -regular
Fiberpreserving
iff q and q|∂N
transverse to f
q : N ⇧M is f -regular i⇤ q and q|�N are transverse to f .
�V = {(x, v) � N ⇥ V | q(x) = f(v)}
(q�(f))(x, v) = x
�q(x, v) = v
q�(f) has the same fiber of f .
2Pullbacks
V
f
MN q
Vq
V
q∗(f)
Pullbackof f
f -regular
Fiberpreserving
iff q and q|∂N
transverse to f
q : N ⇧M is f -regular i⇤ q and q|�N are transverse to f .
�V = {(x, v) � N ⇥ V | q(x) = f(v)}
(q�(f))(x, v) = x
�q(x, v) = v
q�(f) has the same fiber of f .
2Pullbacks
V
f
MN q
Vq
V
q∗(f)
Pullbackof f
f -regular
Fiberpreserving
iff q and q|∂N
transverse to f
q : N ⇧M is f -regular i⇤ q and q|�N are transverse to f .
�V = {(x, v) � N ⇥ V | q(x) = f(v)}
(q�(f))(x, v) = x
�q(x, v) = v
q�(f) has the same fiber of f .
3Universal Lefschetz fibrations
u : U ⇧ M universal ⌃⌥ ⌦ f : V ⇧ N with the same fiber is a pullback ofu for some q : N ⇧M u-regular.
U
u
MN q
Vq
f =q∗(u)
In general we restrict to those f that belong to a given class of Lefschetzfibrations with fiber F : f � L(F ). In this case we talk about L(F )-universality.
In the following theorem, we refer to L(F ) as the class of Lefschetz fibrationsover 2 or 3-manifolds.
3Universal Lefschetz fibrations
u : U ⇧ M universal ⌃⌥ ⌦ f : V ⇧ N with the same fiber is a pullback ofu for some q : N ⇧M u-regular.
U
u
MN q
Vq
f =q∗(u)
In general we restrict to those f that belong to a given class of Lefschetzfibrations with fiber F : f � L(F ). In this case we talk about L(F )-universality.
In the following theorem, we refer to L(F ) as the class of Lefschetz fibrationsover 2 or 3-manifolds.
3Universal Lefschetz fibrations
u : U ⇧ M universal ⌃⌥ ⌦ f : V ⇧ N with the same fiber is a pullback ofu for some q : N ⇧M u-regular.
U
u
MN q
Vq
f =q∗(u)
In general we restrict to those f that belong to a given class of Lefschetzfibrations with fiber F : f � L(F ). In this case we talk about L(F )-universality.
In the following theorem, we refer to L(F ) as the class of Lefschetz fibrationsover 2 or 3-manifolds.
3Universal Lefschetz fibrations
u : U ⇧ M universal ⌃⌥ ⌦ f : V ⇧ N with the same fiber is a pullback ofu for some q : N ⇧M u-regular.
U
u
MN q
Vq
f =q∗(u)
In general we restrict to those f that belong to a given class of Lefschetzfibrations with fiber F : f � L(F ). In this case we talk about L(F )-universality.
In the following theorem, we refer to L(F ) as the class of Lefschetz fibrationsover 2 or 3-manifolds.
3Universal Lefschetz fibrations
u : U ⇧ M universal ⌃⌥ ⌦ f : V ⇧ N with the same fiber is a pullback ofu for some q : N ⇧M u-regular.
U
u
MN q
Vq
f =q∗(u)
In general we restrict to those f that belong to a given class of Lefschetzfibrations with fiber F : f � L(F ). In this case we talk about L(F )-universality.
In the following theorem, we refer to L(F ) as the class of Lefschetz fibrationsover 2 or 3-manifolds.
4Theorem (Z. 2012 & 2014). There exist u2 and u3 that are universal for
genus-g Lefschetz fibrations over 2- and 3-manifolds respectively. Moreover,
universal Lefschetz fibrations can be characterized in terms of monodromy
representations.
Actually, u2 : U62 ⇧M4
2 and u3 : U83 ⇧M6
3 can be constructed explicitly.
One of the the main features for such ui is that the monodromy representation
⇤ui : ⇥1(Mi � Crit(ui))⇧Modg
is an isomorphism, with Modg the mapping class group of genus g. Moreover,our construction gives
M2 = B4 � {2-handles}.
2-handles correspond to relators of a presentation of Modg (having Dehn twistsas generators).
4Theorem (Z. 2012 & 2014). There exist u2 and u3 that are universal for
genus-g Lefschetz fibrations over 2- and 3-manifolds respectively. Moreover,
universal Lefschetz fibrations can be characterized in terms of monodromy
representations.
Actually, u2 : U62 ⇧M4
2 and u3 : U83 ⇧M6
3 can be constructed explicitly.
One of the the main features for such ui is that the monodromy representation
⇤ui : ⇥1(Mi � Crit(ui))⇧Modg
is an isomorphism, with Modg the mapping class group of genus g. Moreover,our construction gives
M2 = B4 � {2-handles}.
2-handles correspond to relators of a presentation of Modg (having Dehn twistsas generators).
4Theorem (Z. 2012 & 2014). There exist u2 and u3 that are universal for
genus-g Lefschetz fibrations over 2- and 3-manifolds respectively. Moreover,
universal Lefschetz fibrations can be characterized in terms of monodromy
representations.
Actually, u2 : U62 ⇧M4
2 and u3 : U83 ⇧M6
3 can be constructed explicitly.
One of the the main features for such ui is that the monodromy representation
⇤ui : ⇥1(Mi � Crit(ui))⇧Modg
is an isomorphism, with Modg the mapping class group of genus g. Moreover,our construction gives
M2 = B4 � {2-handles}.
2-handles correspond to relators of a presentation of Modg (having Dehn twistsas generators).
4Theorem (Z. 2012 & 2014). There exist u2 and u3 that are universal for
genus-g Lefschetz fibrations over 2- and 3-manifolds respectively. Moreover,
universal Lefschetz fibrations can be characterized in terms of monodromy
representations.
Actually, u2 : U62 ⇧M4
2 and u3 : U83 ⇧M6
3 can be constructed explicitly.
One of the the main features for such ui is that the monodromy representation
⇤ui : ⇥1(Mi � Crit(ui))⇧Modg
is an isomorphism, with Modg the mapping class group of genus g. Moreover,our construction gives
M2 = B4 � {2-handles}.
2-handles correspond to relators of a presentation of Modg (having Dehn twistsas generators).
4Theorem (Z. 2012 & 2014). There exist u2 and u3 that are universal for
genus-g Lefschetz fibrations over 2- and 3-manifolds respectively. Moreover,
universal Lefschetz fibrations can be characterized in terms of monodromy
representations.
Actually, u2 : U62 ⇧M4
2 and u3 : U83 ⇧M6
3 can be constructed explicitly.
One of the the main features for such ui is that the monodromy representation
⇤ui : ⇥1(Mi � Crit(ui))⇧Modg
is an isomorphism, with Modg the mapping class group of genus g. Moreover,our construction gives
M2 = B4 � {2-handles}.
2-handles correspond to relators of a presentation of Modg (having Dehn twistsas generators).
4Theorem (Z. 2012 & 2014). There exist u2 and u3 that are universal for
genus-g Lefschetz fibrations over 2- and 3-manifolds respectively. Moreover,
universal Lefschetz fibrations can be characterized in terms of monodromy
representations.
Actually, u2 : U62 ⇧M4
2 and u3 : U83 ⇧M6
3 can be constructed explicitly.
One of the the main features for such ui is that the monodromy representation
⇤ui : ⇥1(Mi � Crit(ui))⇧Modg
is an isomorphism, with Modg the mapping class group of genus g. Moreover,our construction gives
M2 = B4 � {2-handles}.
2-handles correspond to relators of a presentation of Modg (having Dehn twistsas generators).
5Construction of u2 : U6
2 � M42
Assume for simplicity g > 1.
Modg = ��1, . . . , �k | r1, . . . , rl�
�i a Dehn twist.
Consider a Lefschetz fibration v : V ⇧ B2 with fiber Fg having (�1, . . . , �k) asthe monodromy sequence.
δ1 δ2 . . . δk
v is universal for Lefschetz fibrations over surfaces with boundary (Z. 2012).
5Construction of u2 : U6
2 � M42
Assume for simplicity g > 1.
Modg = ��1, . . . , �k | r1, . . . , rl�
�i a Dehn twist.
Consider a Lefschetz fibration v : V ⇧ B2 with fiber Fg having (�1, . . . , �k) asthe monodromy sequence.
δ1 δ2 . . . δk
v is universal for Lefschetz fibrations over surfaces with boundary (Z. 2012).
5Construction of u2 : U6
2 � M42
Assume for simplicity g > 1.
Modg = ��1, . . . , �k | r1, . . . , rl�
�i a Dehn twist.
Consider a Lefschetz fibration v : V ⇧ B2 with fiber Fg having (�1, . . . , �k) asthe monodromy sequence.
δ1 δ2 . . . δk
v is universal for Lefschetz fibrations over surfaces with boundary (Z. 2012).
5Construction of u2 : U6
2 � M42
Assume for simplicity g > 1.
Modg = ��1, . . . , �k | r1, . . . , rl�
�i a Dehn twist.
Consider a Lefschetz fibration v : V ⇧ B2 with fiber Fg having (�1, . . . , �k) asthe monodromy sequence.
δ1 δ2 . . . δk
v is universal for Lefschetz fibrations over surfaces with boundary (Z. 2012).
6Let
v⇥ = id⇥v : B2 ⇥ V ⇧ B2 ⇥B2 ⇤= B4.
The critical image of v⇥ is a collection of parallel trivial 2-disks in B4. So,
⇥1(B4 � Crit(v⇥)) ⇤= ⇥1(S3 � ⇧(Crit(v⇥))) ⇤= ��1, . . . , �k�.
The relators ri’s are words in the �i’s, so they can be represented by pairwisedisjoint embedded loops
ri ⌅ S3 � ⇧(Crit(v⇥)).
The monodromy representation
⇤v� : ⇥1(B4 � Crit(v⇥)) ⇧ Modg
is surjective. We kill the kernel by adding 2-handles H2i to B4 along ri with
an arbitrary framing (for example with framing 0). Let M2 be the resulting4-manifold:
ri � ker(⇤v�) =⌥ v⇥ extends over H2i u2.
6Let
v⇥ = id⇥v : B2 ⇥ V ⇧ B2 ⇥B2 ⇤= B4.
The critical image of v⇥ is a collection of parallel trivial 2-disks in B4. So,
⇥1(B4 � Crit(v⇥)) ⇤= ⇥1(S3 � ⇧(Crit(v⇥))) ⇤= ��1, . . . , �k�.
The relators ri’s are words in the �i’s, so they can be represented by pairwisedisjoint embedded loops
ri ⌅ S3 � ⇧(Crit(v⇥)).
The monodromy representation
⇤v� : ⇥1(B4 � Crit(v⇥)) ⇧ Modg
is surjective. We kill the kernel by adding 2-handles H2i to B4 along ri with
an arbitrary framing (for example with framing 0). Let M2 be the resulting4-manifold:
ri � ker(⇤v�) =⌥ v⇥ extends over H2i u2.
6Let
v⇥ = id⇥v : B2 ⇥ V ⇧ B2 ⇥B2 ⇤= B4.
The critical image of v⇥ is a collection of parallel trivial 2-disks in B4. So,
⇥1(B4 � Crit(v⇥)) ⇤= ⇥1(S3 � ⇧(Crit(v⇥))) ⇤= ��1, . . . , �k�.
The relators ri’s are words in the �i’s, so they can be represented by pairwisedisjoint embedded loops
ri ⌅ S3 � ⇧(Crit(v⇥)).
The monodromy representation
⇤v� : ⇥1(B4 � Crit(v⇥)) ⇧ Modg
is surjective. We kill the kernel by adding 2-handles H2i to B4 along ri with
an arbitrary framing (for example with framing 0). Let M2 be the resulting4-manifold:
ri � ker(⇤v�) =⌥ v⇥ extends over H2i u2.
6Let
v⇥ = id⇥v : B2 ⇥ V ⇧ B2 ⇥B2 ⇤= B4.
The critical image of v⇥ is a collection of parallel trivial 2-disks in B4. So,
⇥1(B4 � Crit(v⇥)) ⇤= ⇥1(S3 � ⇧(Crit(v⇥))) ⇤= ��1, . . . , �k�.
The relators ri’s are words in the �i’s, so they can be represented by pairwisedisjoint embedded loops
ri ⌅ S3 � ⇧(Crit(v⇥)).
The monodromy representation
⇤v� : ⇥1(B4 � Crit(v⇥)) ⇧ Modg
is surjective. We kill the kernel by adding 2-handles H2i to B4 along ri with
an arbitrary framing (for example with framing 0). Let M2 be the resulting4-manifold:
ri � ker(⇤v�) =⌥ v⇥ extends over H2i u2.
6Let
v⇥ = id⇥v : B2 ⇥ V ⇧ B2 ⇥B2 ⇤= B4.
The critical image of v⇥ is a collection of parallel trivial 2-disks in B4. So,
⇥1(B4 � Crit(v⇥)) ⇤= ⇥1(S3 � ⇧(Crit(v⇥))) ⇤= ��1, . . . , �k�.
The relators ri’s are words in the �i’s, so they can be represented by pairwisedisjoint embedded loops
ri ⌅ S3 � ⇧(Crit(v⇥)).
The monodromy representation
⇤v� : ⇥1(B4 � Crit(v⇥)) ⇧ Modg
is surjective. We kill the kernel by adding 2-handles H2i to B4 along ri with
an arbitrary framing (for example with framing 0). Let M2 be the resulting4-manifold:
ri � ker(⇤v�) =⌥ v⇥ extends over H2i u2.
6Let
v⇥ = id⇥v : B2 ⇥ V ⇧ B2 ⇥B2 ⇤= B4.
The critical image of v⇥ is a collection of parallel trivial 2-disks in B4. So,
⇥1(B4 � Crit(v⇥)) ⇤= ⇥1(S3 � ⇧(Crit(v⇥))) ⇤= ��1, . . . , �k�.
The relators ri’s are words in the �i’s, so they can be represented by pairwisedisjoint embedded loops
ri ⌅ S3 � ⇧(Crit(v⇥)).
The monodromy representation
⇤v� : ⇥1(B4 � Crit(v⇥)) ⇧ Modg
is surjective. We kill the kernel by adding 2-handles H2i to B4 along ri with
an arbitrary framing (for example with framing 0). Let M2 be the resulting4-manifold:
ri � ker(⇤v�) =⌥ v⇥ extends over H2i u2.
7Lefschetz cobordism groups
Fix the fiber genus g and the dimension m of the base manifolds.
h : W ⇧ N ⇧h : ⇧W ⇧ ⇧N
f : V ⇧M �f : (�V )⇧ (�M)
f1 + f2 = f1 ✏ f2
f1 : V1 ⇧M1 and f2 : V2 ⇧M2
are cobordant i⇤f1 � f2 = ⇧h
for a Lefschetz fibration h : W ⇧ N .
Let �(g,m) be the set of equivalence classes.It’s an abelian group called the Lefschetz cobordism group.
7Lefschetz cobordism groups
Fix the fiber genus g and the dimension m of the base manifolds.
h : W ⇧ N ⇧h : ⇧W ⇧ ⇧N
f : V ⇧M �f : (�V )⇧ (�M)
f1 + f2 = f1 ✏ f2
f1 : V1 ⇧M1 and f2 : V2 ⇧M2
are cobordant i⇤f1 � f2 = ⇧h
for a Lefschetz fibration h : W ⇧ N .
Let �(g,m) be the set of equivalence classes.It’s an abelian group called the Lefschetz cobordism group.
7Lefschetz cobordism groups
Fix the fiber genus g and the dimension m of the base manifolds.
h : W ⇧ N ⇧h : ⇧W ⇧ ⇧N
f : V ⇧M �f : (�V )⇧ (�M)
f1 + f2 = f1 ✏ f2
f1 : V1 ⇧M1 and f2 : V2 ⇧M2
are cobordant i⇤f1 � f2 = ⇧h
for a Lefschetz fibration h : W ⇧ N .
Let �(g,m) be the set of equivalence classes.It’s an abelian group called the Lefschetz cobordism group.
7Lefschetz cobordism groups
Fix the fiber genus g and the dimension m of the base manifolds.
h : W ⇧ N ⇧h : ⇧W ⇧ ⇧N
f : V ⇧M �f : (�V )⇧ (�M)
f1 + f2 = f1 ✏ f2
f1 : V1 ⇧M1 and f2 : V2 ⇧M2
are cobordant i⇤f1 � f2 = ⇧h
for a Lefschetz fibration h : W ⇧ N .
Let �(g,m) be the set of equivalence classes.It’s an abelian group called the Lefschetz cobordism group.
7Lefschetz cobordism groups
Fix the fiber genus g and the dimension m of the base manifolds.
h : W ⇧ N ⇧h : ⇧W ⇧ ⇧N
f : V ⇧M �f : (�V )⇧ (�M)
f1 + f2 = f1 ✏ f2
f1 : V1 ⇧M1 and f2 : V2 ⇧M2
are cobordant i⇤f1 � f2 = ⇧h
for a Lefschetz fibration h : W ⇧ N .
Let �(g,m) be the set of equivalence classes.It’s an abelian group called the Lefschetz cobordism group.
7Lefschetz cobordism groups
Fix the fiber genus g and the dimension m of the base manifolds.
h : W ⇧ N ⇧h : ⇧W ⇧ ⇧N
f : V ⇧M �f : (�V )⇧ (�M)
f1 + f2 = f1 ✏ f2
f1 : V1 ⇧M1 and f2 : V2 ⇧M2
are cobordant i⇤f1 � f2 = ⇧h
for a Lefschetz fibration h : W ⇧ N .
Let �(g,m) be the set of equivalence classes.It’s an abelian group called the Lefschetz cobordism group.
7Lefschetz cobordism groups
Fix the fiber genus g and the dimension m of the base manifolds.
h : W ⇧ N ⇧h : ⇧W ⇧ ⇧N
f : V ⇧M �f : (�V )⇧ (�M)
f1 + f2 = f1 ✏ f2
f1 : V1 ⇧M1 and f2 : V2 ⇧M2
are cobordant i⇤f1 � f2 = ⇧h
for a Lefschetz fibration h : W ⇧ N .
Let �(g,m) be the set of equivalence classes.It’s an abelian group called the Lefschetz cobordism group.
7Lefschetz cobordism groups
Fix the fiber genus g and the dimension m of the base manifolds.
h : W ⇧ N ⇧h : ⇧W ⇧ ⇧N
f : V ⇧M �f : (�V )⇧ (�M)
f1 + f2 = f1 ✏ f2
f1 : V1 ⇧M1 and f2 : V2 ⇧M2
are cobordant i⇤f1 � f2 = ⇧h
for a Lefschetz fibration h : W ⇧ N .
Let �(g,m) be the set of equivalence classes.It’s an abelian group called the Lefschetz cobordism group.
8A canonical homomorphism
⌦n � N we have a homomorphism induced by f
f : V ⇧M f� : ⇥n(M)⇧ �(g, n)
[q : N ⇧M ] ⇧ [q�(f)]
⇥n(M) is the n-th bordism group of M (a topological invariant), whose ele-ments are bordism classes of maps
q : N ⇧M
with N a closed oriented n-manifold.
q is bordant to q⇥ : N ⇥ ⇧ M ⌃⌥ ↵ a simultaneous extension Q : W ⇧ M
with ⇧W = N ✏ (�N ⇥).
8A canonical homomorphism
⌦n � N we have a homomorphism induced by f
f : V ⇧M f� : ⇥n(M)⇧ �(g, n)
[q : N ⇧M ] ⇧ [q�(f)]
⇥n(M) is the n-th bordism group of M (a topological invariant), whose ele-ments are bordism classes of maps
q : N ⇧M
with N a closed oriented n-manifold.
q is bordant to q⇥ : N ⇥ ⇧ M ⌃⌥ ↵ a simultaneous extension Q : W ⇧ M
with ⇧W = N ✏ (�N ⇥).
8A canonical homomorphism
⌦n � N we have a homomorphism induced by f
f : V ⇧M f� : ⇥n(M)⇧ �(g, n)
[q : N ⇧M ] ⇧ [q�(f)]
⇥n(M) is the n-th bordism group of M (a topological invariant), whose ele-ments are bordism classes of maps
q : N ⇧M
with N a closed oriented n-manifold.
q is bordant to q⇥ : N ⇥ ⇧ M ⌃⌥ ↵ a simultaneous extension Q : W ⇧ M
with ⇧W = N ✏ (�N ⇥).
8A canonical homomorphism
⌦n � N we have a homomorphism induced by f
f : V ⇧M f� : ⇥n(M)⇧ �(g, n)
[q : N ⇧M ] ⇧ [q�(f)]
⇥n(M) is the n-th bordism group of M (a topological invariant), whose ele-ments are bordism classes of maps
q : N ⇧M
with N a closed oriented n-manifold.
q is bordant to q⇥ : N ⇥ ⇧ M ⌃⌥ ↵ a simultaneous extension Q : W ⇧ M
with ⇧W = N ✏ (�N ⇥).
8A canonical homomorphism
⌦n � N we have a homomorphism induced by f
f : V ⇧M f� : ⇥n(M)⇧ �(g, n)
[q : N ⇧M ] ⇧ [q�(f)]
⇥n(M) is the n-th bordism group of M (a topological invariant), whose ele-ments are bordism classes of maps
q : N ⇧M
with N a closed oriented n-manifold.
q is bordant to q⇥ : N ⇥ ⇧ M ⌃⌥ ↵ a simultaneous extension Q : W ⇧ M
with ⇧W = N ✏ (�N ⇥).
8A canonical homomorphism
⌦n � N we have a homomorphism induced by f
f : V ⇧M f� : ⇥n(M)⇧ �(g, n)
[q : N ⇧M ] ⇧ [q�(f)]
⇥n(M) is the n-th bordism group of M (a topological invariant), whose ele-ments are bordism classes of maps
q : N ⇧M
with N a closed oriented n-manifold.
q is bordant to q⇥ : N ⇥ ⇧ M ⌃⌥ ↵ a simultaneous extension Q : W ⇧ M
with ⇧W = N ✏ (�N ⇥).
8A canonical homomorphism
⌦n � N we have a homomorphism induced by f
f : V ⇧M f� : ⇥n(M)⇧ �(g, n)
[q : N ⇧M ] ⇧ [q�(f)]
⇥n(M) is the n-th bordism group of M (a topological invariant), whose ele-ments are bordism classes of maps
q : N ⇧M
with N a closed oriented n-manifold.
q is bordant to q⇥ : N ⇥ ⇧ M ⌃⌥ ↵ a simultaneous extension Q : W ⇧ M
with ⇧W = N ✏ (�N ⇥).
9As an application of universal Lefschetz fibrations we get:
Corollary. u : U ⇧ M universal with respect to Lefschetz fibrations over
n-manifolds =⌥ u� : ⇥n(M)⇧ �(g, n) surjective. So, u2� : ⇥2(M2)⇧ �(g, 2),u3� : ⇥2(M3)⇧ �(g, 2), and u3� : ⇥3(M3)⇧ �(g, 3) are surjective.
Under certain conditions, the natural homomorphism
µ : ⇥n(M)⇧ Hn(M)
[q : N ⇧M ] ⇧ q�([N ])
is an isomorphism.
This is the case for the universal Lefschetz fibration u2 : U2 ⇧M2. That is,
⇥2(M2) ⇤= H2(M2) ⇤= Zk
hence �(g, 2) is a finitely generated abelian group (k = #{relators}).
Consequence: �(g, 2) is generated by the relators in a presentation of Modg.
9As an application of universal Lefschetz fibrations we get:
Corollary. u : U ⇧ M universal with respect to Lefschetz fibrations over
n-manifolds =⌥ u� : ⇥n(M)⇧ �(g, n) surjective. So, u2� : ⇥2(M2)⇧ �(g, 2),u3� : ⇥2(M3)⇧ �(g, 2), and u3� : ⇥3(M3)⇧ �(g, 3) are surjective.
Under certain conditions, the natural homomorphism
µ : ⇥n(M)⇧ Hn(M)
[q : N ⇧M ] ⇧ q�([N ])
is an isomorphism.
This is the case for the universal Lefschetz fibration u2 : U2 ⇧M2. That is,
⇥2(M2) ⇤= H2(M2) ⇤= Zk
hence �(g, 2) is a finitely generated abelian group (k = #{relators}).
Consequence: �(g, 2) is generated by the relators in a presentation of Modg.
9As an application of universal Lefschetz fibrations we get:
Corollary. u : U ⇧ M universal with respect to Lefschetz fibrations over
n-manifolds =⌥ u� : ⇥n(M)⇧ �(g, n) surjective. So, u2� : ⇥2(M2)⇧ �(g, 2),u3� : ⇥2(M3)⇧ �(g, 2), and u3� : ⇥3(M3)⇧ �(g, 3) are surjective.
Under certain conditions, the natural homomorphism
µ : ⇥n(M)⇧ Hn(M)
[q : N ⇧M ] ⇧ q�([N ])
is an isomorphism.
This is the case for the universal Lefschetz fibration u2 : U2 ⇧M2. That is,
⇥2(M2) ⇤= H2(M2) ⇤= Zk
hence �(g, 2) is a finitely generated abelian group (k = #{relators}).
Consequence: �(g, 2) is generated by the relators in a presentation of Modg.
9As an application of universal Lefschetz fibrations we get:
Corollary. u : U ⇧ M universal with respect to Lefschetz fibrations over
n-manifolds =⌥ u� : ⇥n(M)⇧ �(g, n) surjective. So, u2� : ⇥2(M2)⇧ �(g, 2),u3� : ⇥2(M3)⇧ �(g, 2), and u3� : ⇥3(M3)⇧ �(g, 3) are surjective.
Under certain conditions, the natural homomorphism
µ : ⇥n(M)⇧ Hn(M)
[q : N ⇧M ] ⇧ q�([N ])
is an isomorphism.
This is the case for the universal Lefschetz fibration u2 : U2 ⇧M2. That is,
⇥2(M2) ⇤= H2(M2) ⇤= Zk
hence �(g, 2) is a finitely generated abelian group (k = #{relators}).
Consequence: �(g, 2) is generated by the relators in a presentation of Modg.
9As an application of universal Lefschetz fibrations we get:
Corollary. u : U ⇧ M universal with respect to Lefschetz fibrations over
n-manifolds =⌥ u� : ⇥n(M)⇧ �(g, n) surjective. So, u2� : ⇥2(M2)⇧ �(g, 2),u3� : ⇥2(M3)⇧ �(g, 2), and u3� : ⇥3(M3)⇧ �(g, 3) are surjective.
Under certain conditions, the natural homomorphism
µ : ⇥n(M)⇧ Hn(M)
[q : N ⇧M ] ⇧ q�([N ])
is an isomorphism.
This is the case for the universal Lefschetz fibration u2 : U2 ⇧M2. That is,
⇥2(M2) ⇤= H2(M2) ⇤= Zk
hence �(g, 2) is a finitely generated abelian group (k = #{relators}).
Consequence: �(g, 2) is generated by the relators in a presentation of Modg.
10Final comments
Cobordism classes of surface bundles over surfaces are classified by H2(Modg).So, there is a homomorphism
⌅ : H2(Modg)⇧ �(g, 2)
given by considering a bundle as a Lefschetz fibration without critical points.
For g � 3, ⌅ is injective. This follows from the fact that Modg�3 is perfect(Powell 1978).
We haveH2(Modg) ⇤= Z (g � 4)
(Harer 1982 (incorrect) and 1985 for g � 5, later Korkmaz & Stipsicz 2003 forg � 4, based on work of Pitsch 1999).
10Final comments
Cobordism classes of surface bundles over surfaces are classified by H2(Modg).So, there is a homomorphism
⌅ : H2(Modg)⇧ �(g, 2)
given by considering a bundle as a Lefschetz fibration without critical points.
For g � 3, ⌅ is injective. This follows from the fact that Modg�3 is perfect(Powell 1978).
We haveH2(Modg) ⇤= Z (g � 4)
(Harer 1982 (incorrect) and 1985 for g � 5, later Korkmaz & Stipsicz 2003 forg � 4, based on work of Pitsch 1999).
10Final comments
Cobordism classes of surface bundles over surfaces are classified by H2(Modg).So, there is a homomorphism
⌅ : H2(Modg)⇧ �(g, 2)
given by considering a bundle as a Lefschetz fibration without critical points.
For g � 3, ⌅ is injective. This follows from the fact that Modg�3 is perfect(Powell 1978).
We haveH2(Modg) ⇤= Z (g � 4)
(Harer 1982 (incorrect) and 1985 for g � 5, later Korkmaz & Stipsicz 2003 forg � 4, based on work of Pitsch 1999).
10Final comments
Cobordism classes of surface bundles over surfaces are classified by H2(Modg).So, there is a homomorphism
⌅ : H2(Modg)⇧ �(g, 2)
given by considering a bundle as a Lefschetz fibration without critical points.
For g � 3, ⌅ is injective. This follows from the fact that Modg�3 is perfect(Powell 1978).
We haveH2(Modg) ⇤= Z (g � 4)
(Harer 1982 (incorrect) and 1985 for g � 5, later Korkmaz & Stipsicz 2003 forg � 4, based on work of Pitsch 1999).
10Final comments
Cobordism classes of surface bundles over surfaces are classified by H2(Modg).So, there is a homomorphism
⌅ : H2(Modg)⇧ �(g, 2)
given by considering a bundle as a Lefschetz fibration without critical points.
For g � 3, ⌅ is injective. This follows from the fact that Modg�3 is perfect(Powell 1978).
We haveH2(Modg) ⇤= Z (g � 4)
(Harer 1982 (incorrect) and 1985 for g � 5, later Korkmaz & Stipsicz 2003 forg � 4, based on work of Pitsch 1999).
11Questions
(1) How can we compute �(g,m)?
(2) In light of H2(Modg) ⌅ �(g, 2), g � 3, may we consider �(g, 2) as an“enhanced” second homology of Modg?
(3) What properties of Modg reflect to �(g, 2)?
(4) How to construct Lefschetz fibrations that are universal with respect tohigher dimensional base manifolds?
(5) Is there a completion of the universal bundle E Modg ⇧ B Modg whichis a universal Lefschetz fibration (for all base manifolds)?
11Questions
(1) How can we compute �(g,m)?
(2) In light of H2(Modg) ⌅ �(g, 2), g � 3, may we consider �(g, 2) as an“enhanced” second homology of Modg?
(3) What properties of Modg reflect to �(g, 2)?
(4) How to construct Lefschetz fibrations that are universal with respect tohigher dimensional base manifolds?
(5) Is there a completion of the universal bundle E Modg ⇧ B Modg whichis a universal Lefschetz fibration (for all base manifolds)?
11Questions
(1) How can we compute �(g,m)?
(2) In light of H2(Modg) ⌅ �(g, 2), g � 3, may we consider �(g, 2) as an“enhanced” second homology of Modg?
(3) What properties of Modg reflect to �(g, 2)?
(4) How to construct Lefschetz fibrations that are universal with respect tohigher dimensional base manifolds?
(5) Is there a completion of the universal bundle E Modg ⇧ B Modg whichis a universal Lefschetz fibration (for all base manifolds)?
11Questions
(1) How can we compute �(g,m)?
(2) In light of H2(Modg) ⌅ �(g, 2), g � 3, may we consider �(g, 2) as an“enhanced” second homology of Modg?
(3) What properties of Modg reflect to �(g, 2)?
(4) How to construct Lefschetz fibrations that are universal with respect tohigher dimensional base manifolds?
(5) Is there a completion of the universal bundle E Modg ⇧ B Modg whichis a universal Lefschetz fibration (for all base manifolds)?
11Questions
(1) How can we compute �(g,m)?
(2) In light of H2(Modg) ⌅ �(g, 2), g � 3, may we consider �(g, 2) as an“enhanced” second homology of Modg?
(3) What properties of Modg reflect to �(g, 2)?
(4) How to construct Lefschetz fibrations that are universal with respect tohigher dimensional base manifolds?
(5) Is there a completion of the universal bundle E Modg ⇧ B Modg whichis a universal Lefschetz fibration (for all base manifolds)?
11Questions
(1) How can we compute �(g,m)?
(2) In light of H2(Modg) ⌅ �(g, 2), g � 3, may we consider �(g, 2) as an“enhanced” second homology of Modg?
(3) What properties of Modg reflect to �(g, 2)?
(4) How to construct Lefschetz fibrations that are universal with respect tohigher dimensional base manifolds?
(5) Is there a completion of the universal bundle E Modg ⇧ B Modg whichis a universal Lefschetz fibration (for all base manifolds)?