Introduction to Mapping Class Groups - IISER) Bhopalhome.iiserbhopal.ac.in/~kashyap/mapping.pdf · Study of Mod(S) was initiated by Max Dehn and Jakob Nielsen in the 1920’ies. Kashyap

Introduction The mapping class group of the torus T 2 Classification of homeomorphisms

Introduction to Mapping Class GroupsMath seminar - Semester 1, 2011

Kashyap Rajeevsarathy

Department of Mathematics, IISER Bhopal

September 15, 2011

Kashyap Rajeevsarathy Introduction to Mapping Class Groups


What is the Mapping Class Group?




Let S be a closed surface (eg: Torus, Sphere, Klein Bottle etc..).





Let Homeo(S) denote the space of all homeomorphisms on S .





Let Homeo(S) denote the space of all homeomorphisms on S .Homeo(S) is a topological group.






Define an equivalence relation ∼ on Homeo(S) by f ∼ g if andonly if there is path p : [0, 1] → Homeo(S) connecting f and g .







In other words, we are saying that f and g have to be pathhomotopic via points in Homeo(S).







In other words, we are saying that f and g have to be pathhomotopic via points in Homeo(S). Such a homotopy is oftencalled an isotopy.








We define the mapping class group of S denoted by Mod(S) to bethis space Homeo(S)/ ∼.








We define the mapping class group of S denoted by Mod(S) to bethis space Homeo(S)/ ∼. Equivalently, it is the space of pathcomponents or isotopy classes of Homeo(S) denoted byπ0(Homeo(S)).



Examples for orientable sufaces




Note: For orientable surfaces, we often considerMod+(S) = π0(Homeo+(S)) i.e. path components of orientationpreserving homeomorphisms.





Example 1: Mod(S2) = Z2 and Mod+(S2) = 1.





Example 1: Mod(S2) = Z2 and Mod+(S2) = 1.π0(Homeo(S)) = ±1. −1 is the antipodal map which isorientation reversing in even dimensional spheres.






Example 2: Mod(T 2 = S1 × S1) = GL(2,Z) andMod+(T 2 = S1 × S1) = SL(2,Z).






Example 2: Mod(T 2 = S1 × S1) = GL(2,Z) andMod+(T 2 = S1 × S1) = SL(2,Z).



Examples for non-orientable surfaces




Note: Homeo+(S) doesn’t make sense in non-orientable surfaces.





Example 1: Mod(RP2) = 1.





Example 1: Mod(RP2) = 1. Every homeomorphism of RP2 isisotopic to the identity.






Example 2: Mod(Klein Bottle K ) = Z2 ⊕ Z2.






Example 2: Mod(Klein Bottle K ) = Z2 ⊕ Z2.

Example 3: Mod(Genus 3 non-orientable surface N3) = GL(2,Z).



Brief history

Study of Mod(S) was initiated by Max Dehn and JakobNielsen in the 1920’ies.



Brief history


Dehn tried to show existence of a finite set of generators ofMod(S) using its action on circles in S .



Brief history


Dehn tried to show existence of a finite set of generators ofMod(S) using its action on circles in S . He called it anarithmetic field on S .



Brief history



Nielsen tried to classify the individual elements of Mod(S)using methods in hyperbolic geometry.



Brief history




Their work was forgotten for some time until Harveyrediscovered the arithmetic field of Nielsen in 1960’ies.



Brief history




Their work was forgotten for some time until Harveyrediscovered the arithmetic field of Nielsen in 1960’ies. Hemade it into a simplicial complex called it the complex ofcurves(still a widely researched object).



Brief history




Their work was forgotten for some time until Harveyrediscovered the arithmetic field of Nielsen in 1960’ies. Hemade it into a simplicial complex called it the complex ofcurves(still a widely researched object).

Later, William Thurston gave a complete classification ofsurface homeomorphisms and completed the work started byNielsen.



The Nielsen-Thurston classification theorem

Theorem

Every homeomorphism g of S is isotopic to a homeomorphism fsatisfying exactly one of the following:

1 f has finite order.

2 f (C) = C for some collection C of disjoint nonisotopic curvesin S.

3 f is pseudo-Anosov.



The mapping class group of the torus T 2




Recall: Out(G ) = Aut(G )/Inn(G ). Inn(G ) - Conjugate classes ofelements of Aut(G ).





Theorem (Dehn-Nielsen-Baer)

If S is a closed orientable surface of genus g, thenπ0(Homeo(T 2)) ∼= Out(π1(S)).







Recall: π1(T2) = π1(S

1 × S1) = π1(S1)× π1(S

1) = Z× Z.







Recall: π1(T2) = π1(S

1 × S1) = π1(S1)× π1(S

1) = Z× Z.

Therefore, Mod(T 2) ∼= π0(Homeo(T 2)) ∼= Out(Z× Z) =Aut(Z× Z) = GL(2,Z).



Mod+(T 2) = SL(2,Z)



Mod+(T 2) = SL(2,Z)

We know that GL(2,Z) = SL(2,Z) ⊔

[

0 11 0

]

SL(2,Z). Any

A ∈

[

0 11 0

]

SL(2,Z) has det(A) = −1 and therefore switches

orientation of vectors.



Mod+(T 2) = SL(2,Z)


[

0 11 0

]

SL(2,Z). Any

A ∈

[

0 11 0

]



Since T 2 is orientable, we will restrict our attention toMod+(T 2) = SL(2,Z).



Mod+(T 2) = SL(2,Z)


[

0 11 0

]

SL(2,Z). Any

A ∈

[

0 11 0

]



Since T 2 is orientable, we will restrict our attention toMod+(T 2) = SL(2,Z).

π1(T2) = Z

2 = 〈(1, 0), (0, 1)〉.

[

a bc d

]

∈ SL(2,Z) maps the vector

[

10

]

7→

[

ac

]

and

[

01

]

7→

[

bd

]



The commutative diagram

Fact:The fundamental group π1(X ) of a topological space X actson its universal cover X̃ (under specific conditions) via decktransformations.





The quotient X̃/G is homeomorphic to X .





The quotient X̃/G is homeomorphic to X . Therefore, we have thefollowing commutative diagram.

R2

[

a bc d

]

//

/Z2

��

R2

/Z2

��

T 2

h// T 2



Geometric meaning

(1,0) curve or the longitudinal curve

(0,1) curve or the meridional curve

Figure: π1(T2) = Z

2 = 〈(1, 0), (0, 1)〉. The fundamental group of thetorus is generated by the longitudinal and meridional curves.



Geometric meaning

0 0

(1,1)

1

1

11

01

h

(1,0) curve(1,1) curve

Figure: The matrix takes the (1,0)-longitudinal curve to the (1,1) curve.



Geometric meaning

0 0

(1,1)

1

1

11

01

h

(1,0) curve(1,1) curve

Figure: The matrix takes the (1,0)-longitudinal curve to the (1,1) curve.

This is called a Dehn twist (fixes one curve and twists the otheraround the fixed curve).



Classifying the homeomorphisms

Theorem (Nielsen-Thurston)

Every homeomorphism g of S is isotopic to a homeomorphism fsatisfying exactly one of the following:

1 f has finite order.

2 f (C) = C for some collection C of disjoint nonisotopic curvesin S.

3 f is pseudo-Anosov.



Finite order homeomorphisms

Consider A =

[

0 −11 1

]

and B =

[

0 1−1 1

]

.




Consider A =

[

0 −11 1

]

and B =

[

0 1−1 1

]

. A simple calculation

will show that A2 = B3 = −I




Consider A =

[

0 −11 1

]

and B =

[

0 1−1 1

]



Theorem (Serre)

Every finite subgroup of SL(2,Z) is cyclic, generated by an elementconjugate to either Ai or B j .




Consider A =

[

0 −11 1

]

and B =

[

0 1−1 1

]



Theorem (Serre)


Theorem (Bass-Serre)

SL(2,Z) = 〈A,B |A4 = B6 = I ,A2 = B3 = −I 〉 = Z4 ∗Z2Z6 (an

amalgamated product).




Consider A =

[

0 −11 1

]

and B =

[

0 1−1 1

]



Theorem (Serre)


Theorem (Bass-Serre)

SL(2,Z) = 〈A,B |A4 = B6 = I ,A2 = B3 = −I 〉 = Z4 ∗Z2Z6 (an

amalgamated product).

Therefore, all finite order elements have order 2, 3, 4, or 6 and areconjugate to Ai or B j . Note: Trace < 2.



Homeomorphisms that fix a curve or an element of π1(T2)

We have already seen one example - A = [ 1 01 1 ].

11

01

(1,0) curve

(0,1) curve (1,1) curve

Figure: A Dehn twist of (1,0) curve about the (0,1) curve.





11

01

(1,0) curve



In general any such homeomorphism fixes one curve and the twiststhe other c-times about the fixed curve.





11

01

(1,0) curve



In general any such homeomorphism fixes one curve and the twiststhe other c-times about the fixed curve.Therefore, it is conjugateto either

[±1 0c ±1

]

or[±1 c

0 ±1

]

(with the diagonal elements equal).Note: |Trace| = 2.



Pseudo-Anosov homeomorphisms

The term “pseudo-Anosov map” was coined by William Thurston.





He also invented measured foliations when he proved hisclassification of homeomorphisms.






A foliation is a local product structure on the surface.






A foliation is a local product structure on the surface. A measuredfoliation is a foliation equipped with a invariant transverse measureµ.

Definition (Pseuso-Anosov map)

A homeomorphism f of a closed surface S is called pseudo-Anosovif there exists a transverse pair of measured foliations on S, Fs(stable) and Fu (unstable), and a real number λ > 1 such that thefoliations are preserved by f and their transverse measures aremultiplied by 1/λ and λ.




Any A ∈ SL(2,Z) that is pseudo-Anosov has trace(A) > 2.





Fact: If A ∈ SL(2,Z), then the characteristic polynomialdet(A− λI ) = λ2 − trace(A)λ+ 1 = 0.






We will now analyze the pseudo-Anosov map A = [ 2 11 1 ].













The eigen values of this map are λ = 3+√5

2and 1/λ = 3−

√5

2.







The eigen values of this map are λ = 3+√5

2and 1/λ = 3−

√5

2.

The eigen vectors of this map are Vλ =[

1−1+

√

52

]

and

V1/λ =[

1−1−

√

52

]




Since vλ and v1/λ span R2, they induce transverse foliations (or a

product structure) on the torus T 2.






Since Avλ = λvλ, Av1/λ = (1/λ)v1/λ and λ > 0, the leaves of thisfoliation are stretched by a factor λ in the vλ direction






Since Avλ = λvλ, Av1/λ = (1/λ)v1/λ and λ > 0, the leaves of thisfoliation are stretched by a factor λ in the vλ direction and shrunkby a factor 1/λ in the v1/λ direction.







Even though the foliations are preserved, their transverse measuresare multiplied my factors λ and 1/λ.








The Vλ direction is the stable direction as the leaves parallel to Vλ

get closer to each other.








The Vλ direction is the stable direction as the leaves parallel to Vλ

get closer to each other. The V1/λ direction is the unstabledirection as the leaves parallel to V1/λ move farther away fromeach other.



Illustration

A

λ

Stretched by λ

Shrunk my factor 1/

Vλ

V1/ λ

Vλ

V1/ λ


Documents

Introduction to Mapping Class Groups - IISER) Bhopalhome.iiserbhopal.ac.in/~kashyap/mapping.pdf · Study of Mod(S) was initiated by Max Dehn and Jakob Nielsen in the 1920’ies. Kashyap