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 zuddas@kias.re.kr

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)?

Thank you for your attention!!