Concave symplectic embeddings and relations in mappingclass groups of surfaces

Laura Starkston

University of Texas at Austin

December 6, 2014

Laura Starkston (University of Texas at Austin) Concave symplectic embeddings and relations in mapping class groups of surfacesDecember 6, 2014 1 / 15

Contact3-manifolds(M3, ⇠)

x

y

z

Laura Starkston (University of Texas at Austin) Concave symplectic embeddings and relations in mapping class groups of surfacesDecember 6, 2014 2 / 15

Contact3-manifolds(M3, ⇠)

x

y

z

Symplectic convexfillings (X 4,!)

X

MMxI

V

LV! = !

Laura Starkston (University of Texas at Austin) Concave symplectic embeddings and relations in mapping class groups of surfacesDecember 6, 2014 2 / 15

Contact3-manifolds(M3, ⇠)

x

y

z

Open bookdecomposition⇡ : M \ L ! S

1

Symplectic convexfillings (X 4,!)

X

MMxI

V

LV! = !

Laura Starkston (University of Texas at Austin) Concave symplectic embeddings and relations in mapping class groups of surfacesDecember 6, 2014 2 / 15

Contact3-manifolds(M3, ⇠)

x

y

z

Open bookdecomposition⇡ : M \ L ! S

1 ⌃ = ⇡�1(0)“page”

� : ⌃ ! ⌃monodromy

Symplectic convexfillings (X 4,!)

X

MMxI

V

LV! = !

Laura Starkston (University of Texas at Austin) Concave symplectic embeddings and relations in mapping class groups of surfacesDecember 6, 2014 2 / 15

Contact3-manifolds(M3, ⇠)

x

y

z

Open bookdecomposition⇡ : M \ L ! S

1 ⌃ = ⇡�1(0)“page”

� : ⌃ ! ⌃monodromy

Symplectic convexfillings (X 4,!)

X

MMxI

V

LV! = !

Lefschetz fibration

vanishing cyclesc

1

, · · · , cn

Laura Starkston (University of Texas at Austin) Concave symplectic embeddings and relations in mapping class groups of surfacesDecember 6, 2014 2 / 15

Contact3-manifolds(M3, ⇠)

x

y

z

Open bookdecomposition⇡ : M \ L ! S

1 ⌃ = ⇡�1(0)“page”

� : ⌃ ! ⌃monodromy

Symplectic convexfillings (X 4,!)

X

MMxI

V

LV! = !

Lefschetz fibration

vanishing cyclesc

1

, · · · , cn

factorization ofmonodromy

� = Dc1

· · ·Dcn

Right-handedDehn twists

Laura Starkston (University of Texas at Austin) Concave symplectic embeddings and relations in mapping class groups of surfacesDecember 6, 2014 2 / 15

Concave boundaryof ⌫(⌃

1

[ · · ·[⌃n)

+1

Contact3-manifolds(M3, ⇠)

x

y

z

Open bookdecomposition⇡ : M \ L ! S

1 ⌃ = ⇡�1(0)“page”

� : ⌃ ! ⌃monodromy

Symplectic convexfillings (X 4,!)

X

MMxI

V

LV! = !

Lefschetz fibration

vanishing cyclesc

1

, · · · , cn

factorization ofmonodromy

� = Dc1

· · ·Dcn

Right-handedDehn twists

Laura Starkston (University of Texas at Austin) Concave symplectic embeddings and relations in mapping class groups of surfacesDecember 6, 2014 2 / 15

Concave boundaryof ⌫(⌃

1

[ · · ·[⌃n)

+1

Contact3-manifolds(M3, ⇠)

x

y

z

Open bookdecomposition⇡ : M \ L ! S

1 ⌃ = ⇡�1(0)“page”

� : ⌃ ! ⌃monodromy

Complement ofembedded surfacesin CP2 #N CP2

Convex filling

+1

Symplectic convexfillings (X 4,!)

X

MMxI

V

LV! = !

Lefschetz fibration

vanishing cyclesc

1

, · · · , cn

factorization ofmonodromy

� = Dc1

· · ·Dcn

Right-handedDehn twists

Laura Starkston (University of Texas at Austin) Concave symplectic embeddings and relations in mapping class groups of surfacesDecember 6, 2014 2 / 15

Concave Caps and Convex Fillings

+1

-2-1 -1

-2

-2

⌃1

, · · · ,⌃n

surfaces in a 4-manifold intersecting positively transversely

Laura Starkston (University of Texas at Austin) Concave symplectic embeddings and relations in mapping class groups of surfacesDecember 6, 2014 3 / 15

Concave Caps and Convex Fillings

+1

-2-1 -1

-2

-2

⌃1

, · · · ,⌃n

surfaces in a 4-manifold intersecting positively transversely

Theorem (Gay-Stipsicz, Li-Mak)

There exists ! on ⌫(⌃1

[ · · · [ ⌃n) withconvex boundary negative definite

concave boundary with enough b

+

2

Laura Starkston (University of Texas at Austin) Concave symplectic embeddings and relations in mapping class groups of surfacesDecember 6, 2014 3 / 15

Concave Caps and Convex Fillings

+1

-2-1 -1

-2

-2

⌃1

, · · · ,⌃n

surfaces in a 4-manifold intersecting positively transversely

Theorem (Gay-Stipsicz, Li-Mak)

There exists ! on ⌫(⌃1

[ · · · [ ⌃n) withconvex boundary negative definite

concave boundary with enough b

+

2

convexfilling

concave capConvex fillings are more rare than concave caps.

Laura Starkston (University of Texas at Austin) Concave symplectic embeddings and relations in mapping class groups of surfacesDecember 6, 2014 3 / 15

Concave Caps and Convex Fillings

+1

-2-1 -1

-2

-2

⌃1

, · · · ,⌃n

surfaces in a 4-manifold intersecting positively transversely

Theorem (Gay-Stipsicz, Li-Mak)

There exists ! on ⌫(⌃1

[ · · · [ ⌃n) withconvex boundary negative definite

concave boundary with enough b

+

2

convexfilling

concave capConvex fillings are more rare than concave caps.

Idea: Use concave caps to find convex fillings.

Theorem [McDu↵]

A closed symplectic manifold containing a symplecticpositive S

2 is symplectomorphic to CP2 #N CP2.

Laura Starkston (University of Texas at Austin) Concave symplectic embeddings and relations in mapping class groups of surfacesDecember 6, 2014 3 / 15

Concave embedding approach

Theorem (S.)

For a Seifert fibered space Y over S

2

with k singular fibers and e

0

�k � 1, withits canonical contact structure ⇠can:

1

Every convex filling of (Y , ⇠can) is the complement of a

symplectic embedding of a concave star-shaped plumbing of

spheres into CP2 #N CP2

.

+1

-2-1 -1

-2

-2

Laura Starkston (University of Texas at Austin) Concave symplectic embeddings and relations in mapping class groups of surfacesDecember 6, 2014 4 / 15

Concave embedding approach

Theorem (S.)

For a Seifert fibered space Y over S

2

with k singular fibers and e

0

�k � 1, withits canonical contact structure ⇠can:

1

Every convex filling of (Y , ⇠can) is the complement of a

symplectic embedding of a concave star-shaped plumbing of

spheres into CP2 #N CP2

.

2

Every such embedding can be built from

A collection of pseudoholomorphic CP1

’s

+1

-2-1 -1

-2

-2

Laura Starkston (University of Texas at Austin) Concave symplectic embeddings and relations in mapping class groups of surfacesDecember 6, 2014 4 / 15

Concave embedding approach

Theorem (S.)

For a Seifert fibered space Y over S

2

with k singular fibers and e

0

�k � 1, withits canonical contact structure ⇠can:

1

Every convex filling of (Y , ⇠can) is the complement of a

symplectic embedding of a concave star-shaped plumbing of

spheres into CP2 #N CP2

.

2

Every such embedding can be built from

A collection of pseudoholomorphic CP1

’s

blow-up at N points, including proper transforms of the

CP1

’s and exceptional spheres into the concave plumbing

+1

-2-1 -1

-2

-2

1-1

-2

-1-1-1

-1 -1-1

1-2

-2

-1-1-1

-1 -1-1

-2

Laura Starkston (University of Texas at Austin) Concave symplectic embeddings and relations in mapping class groups of surfacesDecember 6, 2014 4 / 15

Concave embedding approach

Theorem (S.)

For a Seifert fibered space Y over S

2

with k singular fibers and e

0

�k � 1, withits canonical contact structure ⇠can:

1

Every convex filling of (Y , ⇠can) is the complement of a

symplectic embedding of a concave star-shaped plumbing of

spheres into CP2 #N CP2

.

2

Every such embedding can be built from

A collection of pseudoholomorphic CP1

’s

blow-up at N points, including proper transforms of the

CP1

’s and exceptional spheres into the concave plumbing

3

For many such (Y , ⇠can) the isotopy class of the embedding

is determined by combinatorial/homological data

(su�cient conditions: k 5 or e

0

�k � 3).

+1

-2-1 -1

-2

-2

1-1

-2

-1-1-1

-1 -1-1

1-2

-2

-1-1-1

-1 -1-1

-2

Laura Starkston (University of Texas at Austin) Concave symplectic embeddings and relations in mapping class groups of surfacesDecember 6, 2014 4 / 15

Monodromy factorization approach

[Gay-Mark] For these Seifert fibered spaces, (Y , ⇠can), there are open bookdecompositions with

Planar pages

Monodromy � = Dc1

Dc2

· · ·Dcn ,c

1

, · · · , cn disjoint

Theorem (Wendl)

Because these contact structures are planar, each (minimal) convex filling of

(Y , ⇠can) corresponds to a di↵erent positive factorization of �.

Laura Starkston (University of Texas at Austin) Concave symplectic embeddings and relations in mapping class groups of surfacesDecember 6, 2014 5 / 15

Comparing and translating the two methods

Monodromy substitution approach

Easy to write down

Good for applications for cut and paste operations

Concave embedding approach

Easy to find interesting examples

Easy (in many cases) to classify all fillings of certain contact manifolds(combinatorial reduction)

Translating between the two approaches: Handlebody decompositions forembedded surfaces and Lefschetz fibrations

Laura Starkston (University of Texas at Austin) Concave symplectic embeddings and relations in mapping class groups of surfacesDecember 6, 2014 6 / 15

How to draw an embedding of spheres in a 4-manifold

Unknotted circles in S

3 $ equators of spheresFraming $ self-intersection numberLinking $ Pairwise intersections

+1

+1 0 0-1

Laura Starkston (University of Texas at Austin) Concave symplectic embeddings and relations in mapping class groups of surfacesDecember 6, 2014 7 / 15

Blowing up and down

-1

+1

-1

Laura Starkston (University of Texas at Austin) Concave symplectic embeddings and relations in mapping class groups of surfacesDecember 6, 2014 8 / 15

How to draw a planar Lefchetz fibration

Dotted circle notation: Attachinga 1-handle is equivalent to carving out a 2-handle– a dotted circle denotes the boundary of this core disk.

1

2

34

5

6

7c1

c2

c3

12 3 4 5 6 7

c1c2

c3

Laura Starkston (University of Texas at Austin) Concave symplectic embeddings and relations in mapping class groups of surfacesDecember 6, 2014 9 / 15

+1

-n1 -n2 -nd

Concave embeddings: Start with d + 1 pseudoholomorphic CP1’s in CP2.

+1

CP2

Laura Starkston (University of Texas at Austin) Concave symplectic embeddings and relations in mapping class groups of surfacesDecember 6, 2014 10 / 15

+1

-1 -1-1

Concave embeddings: Start with 3 + 1 pseudoholomorphic CP1’s in CP2.

+1

0

0

0

cancelling 3-handles

CP2

Laura Starkston (University of Texas at Austin) Concave symplectic embeddings and relations in mapping class groups of surfacesDecember 6, 2014 10 / 15

+1

-1 -1-1

Concave embeddings: Start with 3 + 1 pseudoholomorphic CP1’s in CP2.

+1

cancelling 3-handles

+1

+1+1 +1

CP2

Laura Starkston (University of Texas at Austin) Concave symplectic embeddings and relations in mapping class groups of surfacesDecember 6, 2014 10 / 15

+1

-1 -1-1

Concave embeddings: Start with 3 + 1 pseudoholomorphic CP1’s in CP2.

+1

cancelling 3-handles

+1 +1 +1

CP2

Laura Starkston (University of Texas at Austin) Concave symplectic embeddings and relations in mapping class groups of surfacesDecember 6, 2014 10 / 15

+1

-1 -1-1

Concave embeddings: Start with 3 + 1 pseudoholomorphic CP1’s in CP2.

+1

cancelling 3-handles

+1 +1 +1

+1

cancelling 3-handles

+1 0 0-1

Blow up to remove intersections and change framings. CP2 #CP2

Laura Starkston (University of Texas at Austin) Concave symplectic embeddings and relations in mapping class groups of surfacesDecember 6, 2014 10 / 15

+1

-1 -1-1

Concave embeddings: Start with 3 + 1 pseudoholomorphic CP1’s in CP2.

+1

cancelling 3-handles

+1 +1 +1

+1

cancelling 3-handles

0 0 -1-1

-1

Blow up to remove intersections and change framings. CP2 #2CP2

Laura Starkston (University of Texas at Austin) Concave symplectic embeddings and relations in mapping class groups of surfacesDecember 6, 2014 10 / 15

+1

-1 -1-1

Concave embeddings: Start with 3 + 1 pseudoholomorphic CP1’s in CP2.

+1

cancelling 3-handles

+1 +1 +1

+1

cancelling 3-handles

-1 -1 -1-1

-1

-1

Blow up to remove intersections and change framings. CP2 #3CP2

Laura Starkston (University of Texas at Austin) Concave symplectic embeddings and relations in mapping class groups of surfacesDecember 6, 2014 10 / 15

+1

-1 -1-1

Concave embeddings: Start with 3 + 1 pseudoholomorphic CP1’s in CP2.

+1

cancelling 3-handles

-1 -1 -1-1

-1

-1

+1

cancelling 3-handles

-1 -1 -1-1

-1

-1

Notice the embedding of the dual graph into CP2 #3CP2

Laura Starkston (University of Texas at Austin) Concave symplectic embeddings and relations in mapping class groups of surfacesDecember 6, 2014 10 / 15

+1

-1 -1-1

Concave embeddings: Start with 3 + 1 pseudoholomorphic CP1’s in CP2.

+1

cancelling 3-handles

-1 -1 -1-1

-1

-1

-1

-1

-1

Cut out concave cap from CP2 #3CP2. Turn upsidedown. Simplify

Laura Starkston (University of Texas at Austin) Concave symplectic embeddings and relations in mapping class groups of surfacesDecember 6, 2014 10 / 15

+1

-1 -1-1

Concave embeddings: Start with 3 + 1 pseudoholomorphic CP1’s in CP2.

-1

-1

-1-1

-1

-1

Lefschetz Fibration!

Laura Starkston (University of Texas at Austin) Concave symplectic embeddings and relations in mapping class groups of surfacesDecember 6, 2014 10 / 15

+1

-1 -1-1

Concave embeddings: Start with 3 + 1 pseudoholomorphic CP1’s in CP2.

-1-1

-1

Lantern Relation! D

1

D

2

D

3

D

1,2,3 = D

1,2D1,3D2,3

Laura Starkston (University of Texas at Austin) Concave symplectic embeddings and relations in mapping class groups of surfacesDecember 6, 2014 10 / 15

The lantern relation concave embeddingcame from blowing up:

By blowing up more interestingconfigurations of lines, we get newrelations:

11

11

D

2

1

D

2

2

D

3

D

2

4

D

2

5

D

1,2,3,4,5 = D

1,2,3D1,4D1,5D2,4D2,5D3,4,5

Laura Starkston (University of Texas at Austin) Concave symplectic embeddings and relations in mapping class groups of surfacesDecember 6, 2014 11 / 15

D

2

1

D

2

2

D

3

D

2

4

D

2

5

D

1,2,3,4,5 = D

1,2,3D1,4D1,5D2,4D2,5D3,4,5

Concave embedding strategy shows: no other fillings ) no other + factorizations.

Such indecomposable relations are essential relators for elements in Dehn+.

There is an infinite family of indecomposable relations generalizing this example.

Laura Starkston (University of Texas at Austin) Concave symplectic embeddings and relations in mapping class groups of surfacesDecember 6, 2014 11 / 15

Longer Arms

+1

-2 -2 -2

+1

-2 -2 -2

-2

+1

cancelling 3-handles

-2 -2 -2-1

-1-1

-1-1-1-1-2

-1

Laura Starkston (University of Texas at Austin) Concave symplectic embeddings and relations in mapping class groups of surfacesDecember 6, 2014 12 / 15

Longer Arms

+1

-1 -1-1

+1

-2 -2 -2

-2

+1

cancelling 3-handles

-2 -1 -2-1

-1-1

-1 -1 -1 -2

-1

Laura Starkston (University of Texas at Austin) Concave symplectic embeddings and relations in mapping class groups of surfacesDecember 6, 2014 12 / 15

-1-2

-2 -1-1

-1-2

-2 -1-1

Laura Starkston (University of Texas at Austin) Concave symplectic embeddings and relations in mapping class groups of surfacesDecember 6, 2014 13 / 15

-1-2

-2 -1-1

-1-2

-2 -1-1

-1

-1 -2-1

-1

-2

Laura Starkston (University of Texas at Austin) Concave symplectic embeddings and relations in mapping class groups of surfacesDecember 6, 2014 13 / 15

D

1

D

2

D

3

D

1,2,3 = D

1,2D1,3D2,3

D

1aD1bD1a1bD2

2

D

3

D

1a,1b,2,3 = D

1a,2D1b,2D1a,1b,3D2,3

Laura Starkston (University of Texas at Austin) Concave symplectic embeddings and relations in mapping class groups of surfacesDecember 6, 2014 14 / 15

Moving towards a complete dictionary: other moves

What is a complete list of embedding moves, and how do they each translateto moves on mapping class group relations?

How does a sequence of embedding moves translate to a sequence ofmapping class group relations?

a b a-1 b-1

-1

a-1 b-2

-1-2

a-2 b-2

-1 -2-3

-1

a ba-1 b-1

-1

a-1 b-2

-1-3

cc-1 c-2

-1

Laura Starkston (University of Texas at Austin) Concave symplectic embeddings and relations in mapping class groups of surfacesDecember 6, 2014 15 / 15