Non commutative representations of Torelli groups Christian - - PowerPoint PPT Presentation

Homological representations Surface braid groups Heisenberg groups Regular Torelli groups

Non commutative representations of Torelli groups

Christian Blanchet, Univ. Paris Diderot, IMJ Swiss Knots 2011 Lake Thun, May 2011

Homological representations Surface braid groups Heisenberg groups Regular Torelli groups

Homological representations of configuration spaces Surface braid groups Heisenberg groups and covers Representations of regular Torelli groups

Homological representations Surface braid groups Heisenberg groups Regular Torelli groups

Configurations in the disc

• Cn(D2

m) =

• (D2 − {z1, . . . , zm})n − Diag
• /Sn,
• Cn(D2

m) is a regular covering associated with a quotient map

ψ : π1(Cn(D2

m), ∗) → Z2 =< q, t > ,

defined using abelianisation of Bn+m and Bm.

Homological representations Surface braid groups Heisenberg groups Regular Torelli groups

Configurations in the disc

• Cn(D2

m) =

• (D2 − {z1, . . . , zm})n − Diag
• /Sn,
• Cn(D2

m) is a regular covering associated with a quotient map

ψ : π1(Cn(D2

m), ∗) → Z2 =< q, t > ,

defined using abelianisation of Bn+m and Bm.

• H∗(

Cn(D2

m)) is a Z[q±1, t±1]-module.

Homological representations Surface braid groups Heisenberg groups Regular Torelli groups

Configurations in the disc

• Cn(D2

m) =

• (D2 − {z1, . . . , zm})n − Diag
• /Sn,
• Cn(D2

m) is a regular covering associated with a quotient map

ψ : π1(Cn(D2

m), ∗) → Z2 =< q, t > ,

defined using abelianisation of Bn+m and Bm.

• H∗(

Cn(D2

m)) is a Z[q±1, t±1]-module.

• For f ∈ Diff(D2

m, S1), F = Cn(f ) lifts to

Cn(D2

m) .

Homological representations Surface braid groups Heisenberg groups Regular Torelli groups

Configurations in the disc

• Cn(D2

m) =

• (D2 − {z1, . . . , zm})n − Diag
• /Sn,
• Cn(D2

m) is a regular covering associated with a quotient map

ψ : π1(Cn(D2

m), ∗) → Z2 =< q, t > ,

defined using abelianisation of Bn+m and Bm.

• H∗(

Cn(D2

m)) is a Z[q±1, t±1]-module.

• For f ∈ Diff(D2

m, S1), F = Cn(f ) lifts to

Cn(D2

m) .

F∗ : H∗( Cn(D2

m)) → H∗(

Cn(D2

m)) is Z[q±1, t±1]-linear .

Homological representations Surface braid groups Heisenberg groups Regular Torelli groups

Configurations in the disc

• Cn(D2

m) =

• (D2 − {z1, . . . , zm})n − Diag
• /Sn,
• Cn(D2

m) is a regular covering associated with a quotient map

ψ : π1(Cn(D2

m), ∗) → Z2 =< q, t > ,

defined using abelianisation of Bn+m and Bm.

• H∗(

Cn(D2

m)) is a Z[q±1, t±1]-module.

• For f ∈ Diff(D2

m, S1), F = Cn(f ) lifts to

Cn(D2

m) .

F∗ : H∗( Cn(D2

m)) → H∗(

Cn(D2

m)) is Z[q±1, t±1]-linear .

• This defines a representation of the braid group

Bm ≈ MCG(D2

m, S1) acting on Hn(

Cn(D2

m)) [Lawrence,

Krammer (n = 2), Bigelow].

Homological representations Surface braid groups Heisenberg groups Regular Torelli groups

Configurations in the disc

• Cn(D2

m) =

• (D2 − {z1, . . . , zm})n − Diag
• /Sn,
• Cn(D2

m) is a regular covering associated with a quotient map

ψ : π1(Cn(D2

m), ∗) → Z2 =< q, t > ,

defined using abelianisation of Bn+m and Bm.

• H∗(

Cn(D2

m)) is a Z[q±1, t±1]-module.

• For f ∈ Diff(D2

m, S1), F = Cn(f ) lifts to

Cn(D2

m) .

F∗ : H∗( Cn(D2

m)) → H∗(

Cn(D2

m)) is Z[q±1, t±1]-linear .

• This defines a representation of the braid group

Bm ≈ MCG(D2

m, S1) acting on Hn(

Cn(D2

m)) [Lawrence,

Krammer (n = 2), Bigelow].

• Variants: action on Hn(

Cn(D2

m), Hn(

Cn(D2

m)), ∂

Cn(D2

m))),

Hn( Cn(D2

m)),

νǫ), . . .

Homological representations Surface braid groups Heisenberg groups Regular Torelli groups

Moriyama representations

• Moriyama (J. London Math. 2007) considers the action of the

MCG of Σ = Σg,1 on homology of the space of ordered configurations.

Homological representations Surface braid groups Heisenberg groups Regular Torelli groups

Moriyama representations

• Moriyama (J. London Math. 2007) considers the action of the

MCG of Σ = Σg,1 on homology of the space of ordered configurations.

• This reproduces the Johnson filtration. Let p0 ∈ ∂Σ, n > 0.

Theorem (Moriyama) The kernel of the action of MCG on Hn(Σn, diag ∪ {contains p0}) is the n-th Torelli subgroup Tn(Σ).

Homological representations Surface braid groups Heisenberg groups Regular Torelli groups

Moriyama representations

• Moriyama (J. London Math. 2007) considers the action of the

MCG of Σ = Σg,1 on homology of the space of ordered configurations.

• This reproduces the Johnson filtration. Let p0 ∈ ∂Σ, n > 0.

Theorem (Moriyama) The kernel of the action of MCG on Hn(Σn, diag ∪ {contains p0}) is the n-th Torelli subgroup Tn(Σ).

• The kernel of the action on

Hn((Σn, diag ∪ {contains p0})/Sn) is the Torelli group T1(Σ).

Homological representations Surface braid groups Heisenberg groups Regular Torelli groups

Moriyama representations

• Moriyama (J. London Math. 2007) considers the action of the

MCG of Σ = Σg,1 on homology of the space of ordered configurations.

• This reproduces the Johnson filtration. Let p0 ∈ ∂Σ, n > 0.

Theorem (Moriyama) The kernel of the action of MCG on Hn(Σn, diag ∪ {contains p0}) is the n-th Torelli subgroup Tn(Σ).

• The kernel of the action on

Hn((Σn, diag ∪ {contains p0})/Sn) is the Torelli group T1(Σ).

• Nice cell decomposition of the pair

(Σn, diag ∪ {contains p0}) compatible with Sn action.

Homological representations Surface braid groups Heisenberg groups Regular Torelli groups

Ar and Ko extension of Lawrence-Krammer-Bigelow representations

• Ar and Ko [Pacific J. 2010] have extended

Lawrence-Krammer-Bigelow representations to surface braid groups.

Homological representations Surface braid groups Heisenberg groups Regular Torelli groups

Ar and Ko extension of Lawrence-Krammer-Bigelow representations

• Ar and Ko [Pacific J. 2010] have extended

Lawrence-Krammer-Bigelow representations to surface braid groups.

• Our construction is similar.

Our homologies have different rank and we focus on action of mapping classes.

Homological representations Surface braid groups Heisenberg groups Regular Torelli groups

Homological representations of configuration spaces Surface braid groups Heisenberg groups and covers Representations of regular Torelli groups

Homological representations Surface braid groups Heisenberg groups Regular Torelli groups

Surface braids

• Σ = Σg,1 is an oriented genus g surface with one boundary

component; Cn(Σ) = (Σn − Diag)/Sn, Bn(Σ) = π1(Cn(Σ), ∗).

Homological representations Surface braid groups Heisenberg groups Regular Torelli groups

Surface braids

• Σ = Σg,1 is an oriented genus g surface with one boundary

component; Cn(Σ) = (Σn − Diag)/Sn, Bn(Σ) = π1(Cn(Σ), ∗).

• Σ = D2/ ∼,

where ∼ identifies edges according to the word c g

i=1 biaibiai.

A loop in Cn(Σ) is represented by its graph in [0, 1] × D2.

Homological representations Surface braid groups Heisenberg groups Regular Torelli groups

Presentation of surface braid groups

• P. Bellingeri 2004.

Closed case: G.P. Scott 1970, J. Gonzales Meneses 2001.

• Generators: σ1, . . . , σn−1, a1, . . . , ag, b1, . . . , bg .
• Relations: usual braid relations on σi and mixed relations

below. (R1) arσi = σiar (1 ≤ r ≤ g; i = 1) ; brσi = σibr (1 ≤ r ≤ g; i = 1) ; (R2) σ1arσ1ar = arσ1arσ1 (1 ≤ r ≤ g) ; σ1brσ1br = brσ1brσ1 (1 ≤ r ≤ g) ; (R3) σ−1

1 asσ1ar = arσ−1 1 asσ1

(s < r) ; σ−1

1 bsσ1br = brσ−1 1 bsσ1

(s < r) ; σ−1

1 asσ1br = brσ−1 1 asσ1

(s < r) ; σ−1

1 bsσ1ar = arσ−1 1 bsσ1

(s < r) ; (R4) σ−1

1 arσ1br = brσ1arσ1

(1 ≤ r ≤ g) .

Homological representations Surface braid groups Heisenberg groups Regular Torelli groups

Homological representations of configuration spaces Surface braid groups Heisenberg groups and covers Representations of regular Torelli groups

Homological representations Surface braid groups Heisenberg groups Regular Torelli groups

Central extension of H = H1(Σ)

H = Z × H with (k, h)(k′, h′) = (k + k′ + h.h′, h + h′).

Homological representations Surface braid groups Heisenberg groups Regular Torelli groups

Central extension of H = H1(Σ)

H = Z × H with (k, h)(k′, h′) = (k + k′ + h.h′, h + h′).

H ≃   1 X k Ig Y 1   ⊂ SLg+2(Z).

Homological representations Surface braid groups Heisenberg groups Regular Torelli groups

Central extension of H = H1(Σ)

H = Z × H with (k, h)(k′, h′) = (k + k′ + h.h′, h + h′).

H ≃   1 X k Ig Y 1   ⊂ SLg+2(Z).

• Generators: u, Ai, Bi, 1 ≤ i ≤ g.

Relations: u central, AiBi = u2BiAi (1 ≤ i ≤ g), AiAj = AjAi (1 ≤ i < j ≤ g), AiBj = BjAi (1 ≤ i = j ≤ g).

Homological representations Surface braid groups Heisenberg groups Regular Torelli groups

Heisenberg quotient of surface braid group

• For n ≥ 2, there exists a well defined quotient map

φ : Bn(Σ) → H , such that φ(σk) = u (1 ≤ k ≤ n − 1), φ(ai) = Ai, φ(bi) = Bi (1 ≤ i ≤ g).

Homological representations Surface braid groups Heisenberg groups Regular Torelli groups

Heisenberg quotient of surface braid group

• For n ≥ 2, there exists a well defined quotient map

φ : Bn(Σ) → H , such that φ(σk) = u (1 ≤ k ≤ n − 1), φ(ai) = Ai, φ(bi) = Bi (1 ≤ i ≤ g).

• Bellingeri observation: For n ≥ 3, this quotient coincides with

the second nilpotent quotient Bn(Σ)/Γ3(Bn(Σ)) studied by Bellingeri-Gervais-Guaschi [J. Algebra, 2008].

Homological representations Surface braid groups Heisenberg groups Regular Torelli groups

Heisenberg cover of configuration spaces

Cn(Σ) is the regular cover with group H associated with the quotient map φ : Bn(Σ) → H.

Homological representations Surface braid groups Heisenberg groups Regular Torelli groups

Heisenberg cover of configuration spaces

Cn(Σ) is the regular cover with group H associated with the quotient map φ : Bn(Σ) → H.

• H∗(

Cn(Σ)) is a right Z[ H]-module.

Homological representations Surface braid groups Heisenberg groups Regular Torelli groups

Homological representations of configuration spaces Surface braid groups Heisenberg groups and covers Representations of regular Torelli groups

Homological representations Surface braid groups Heisenberg groups Regular Torelli groups

Action of a mapping class

• Let f be a diffeomorphism of Σ which is identity on a collar of

the boundary containing the base point in Cn(Σ), and F = Cn(f ) : Cn(Σ) → Cn(Σ). F♯ : Bn(Σ) → Bn(Σ) respects the kernel of the quotient map φ : Bn(Σ) → H.

Homological representations Surface braid groups Heisenberg groups Regular Torelli groups

Action of a mapping class

• Let f be a diffeomorphism of Σ which is identity on a collar of

the boundary containing the base point in Cn(Σ), and F = Cn(f ) : Cn(Σ) → Cn(Σ). F♯ : Bn(Σ) → Bn(Σ) respects the kernel of the quotient map φ : Bn(Σ) → H.

• There is a lift

F : Cn(Σ) → Cn(Σ).

Homological representations Surface braid groups Heisenberg groups Regular Torelli groups

Action of a mapping class

• Let f be a diffeomorphism of Σ which is identity on a collar of

the boundary containing the base point in Cn(Σ), and F = Cn(f ) : Cn(Σ) → Cn(Σ). F♯ : Bn(Σ) → Bn(Σ) respects the kernel of the quotient map φ : Bn(Σ) → H.

• There is a lift

F : Cn(Σ) → Cn(Σ).

• For λ ∈ Z[

H], x ∈ H∗( Cn(Σ)), one has

• F∗(xλ) =

F∗(x)f∗(λ) . Here we denote by f∗ the natural action of f on Z[ H].

Homological representations Surface braid groups Heisenberg groups Regular Torelli groups

Action of a Torelli class

• Suppose that f represents an element of the Torelli group,

then for h ∈ H: f∗(h) = uk(h)h . Theorem a) kf induces a well defined homomorphism H → Z. b) kf is trivial if and only if f acts trivially on homotopy classes of non singular vector fields.

Homological representations Surface braid groups Heisenberg groups Regular Torelli groups

Action of a Torelli class

• Suppose that f represents an element of the Torelli group,

then for h ∈ H: f∗(h) = uk(h)h . Theorem a) kf induces a well defined homomorphism H → Z. b) kf is trivial if and only if f acts trivially on homotopy classes of non singular vector fields.

• The homomorphism f → kf ∈ Hom(H, Z) ≈ H1(Σ) coincides

with the Chillingworth homomorphism.

Homological representations Surface braid groups Heisenberg groups Regular Torelli groups

Action of a Torelli class

• Suppose that f represents an element of the Torelli group,

then for h ∈ H: f∗(h) = uk(h)h . Theorem a) kf induces a well defined homomorphism H → Z. b) kf is trivial if and only if f acts trivially on homotopy classes of non singular vector fields.

• The homomorphism f → kf ∈ Hom(H, Z) ≈ H1(Σ) coincides

with the Chillingworth homomorphism.

• It is trivial on the Johnson subgroup, generated by Dehn

twists along bounding curves.

Homological representations Surface braid groups Heisenberg groups Regular Torelli groups

Regular Torelli group

• We denote by T R(Σ) the regular Torelli group , formed with

those mapping classes acting trivially on homotopy classes of non singular vector fields. Theorem For n ≥ 2, there exists a Z[ H]-linear action of the regular Torelli group T R(Σ) on H∗( Cn(Σ) and its variants.

Homological representations Surface braid groups Heisenberg groups Regular Torelli groups

Description of some cycles, n = 2 case

• If α, α′ are disjoint properly embedded oriented arcs, then a

lift of [α × α′] is a relative 2-cycle representing a class in H2( C2(Σ), ∂ C2(Σ)). A lifting is fixed by connecting α, α′ to the base points.

Homological representations Surface braid groups Heisenberg groups Regular Torelli groups

Description of some cycles, n = 2 case

• If α, α′ are disjoint properly embedded oriented arcs, then a

lift of [α × α′] is a relative 2-cycle representing a class in H2( C2(Σ), ∂ C2(Σ)). A lifting is fixed by connecting α, α′ to the base points.

• Let xo be a point in ∂σ, and Σ′ = Σ − {x0}.

If γ is an embedded loop at xo, then a lift of C2(γ) represents a class in HBM

2

( C2(Σ′)) := H2( C2(Σ′), νǫ).

Homological representations Surface braid groups Heisenberg groups Regular Torelli groups

Intersection pairing

• There exists an intersection pairing

H2( C2(Σ′), ∂ C2(Σ′)) ⊗ HBM

2

( C2(Σ′)) → Z[ H] .

Homological representations Surface braid groups Heisenberg groups Regular Torelli groups

Intersection pairing

• There exists an intersection pairing

H2( C2(Σ′), ∂ C2(Σ′)) ⊗ HBM

2

( C2(Σ′)) → Z[ H] .

• Theorem

The above pairing is non singular, and the two Z[ H]-modules above are free of rank g(2g + 1).