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 � � ( D 2 − { z 1 , . . . , z m } ) n − Diag • C n ( D 2 m ) = / S n , � C n ( D 2 m ) is a regular covering associated with a quotient map m ) , ∗ ) → Z 2 = < q , t > , ψ : π 1 ( C n ( D 2 defined using abelianisation of B n + m and B m .

Homological representations Surface braid groups Heisenberg groups Regular Torelli groups Configurations in the disc � � ( D 2 − { z 1 , . . . , z m } ) n − Diag • C n ( D 2 m ) = / S n , � C n ( D 2 m ) is a regular covering associated with a quotient map m ) , ∗ ) → Z 2 = < q , t > , ψ : π 1 ( C n ( D 2 defined using abelianisation of B n + m and B m . • H ∗ ( � C n ( D 2 m )) is a Z [ q ± 1 , t ± 1 ]-module.

Homological representations Surface braid groups Heisenberg groups Regular Torelli groups Configurations in the disc � � ( D 2 − { z 1 , . . . , z m } ) n − Diag • C n ( D 2 m ) = / S n , � C n ( D 2 m ) is a regular covering associated with a quotient map m ) , ∗ ) → Z 2 = < q , t > , ψ : π 1 ( C n ( D 2 defined using abelianisation of B n + m and B m . • H ∗ ( � C n ( D 2 m )) is a Z [ q ± 1 , t ± 1 ]-module. m , S 1 ), F = C n ( f ) lifts to � • For f ∈ Diff( D 2 C n ( D 2 m ) .

Homological representations Surface braid groups Heisenberg groups Regular Torelli groups Configurations in the disc � � ( D 2 − { z 1 , . . . , z m } ) n − Diag • C n ( D 2 m ) = / S n , � C n ( D 2 m ) is a regular covering associated with a quotient map m ) , ∗ ) → Z 2 = < q , t > , ψ : π 1 ( C n ( D 2 defined using abelianisation of B n + m and B m . • H ∗ ( � C n ( D 2 m )) is a Z [ q ± 1 , t ± 1 ]-module. m , S 1 ), F = C n ( f ) lifts to � • For f ∈ Diff( D 2 C n ( D 2 m ) . • � F ∗ : H ∗ ( � m )) → H ∗ ( � C n ( D 2 C n ( D 2 m )) is Z [ q ± 1 , t ± 1 ]-linear .

Homological representations Surface braid groups Heisenberg groups Regular Torelli groups Configurations in the disc � � ( D 2 − { z 1 , . . . , z m } ) n − Diag • C n ( D 2 m ) = / S n , � C n ( D 2 m ) is a regular covering associated with a quotient map m ) , ∗ ) → Z 2 = < q , t > , ψ : π 1 ( C n ( D 2 defined using abelianisation of B n + m and B m . • H ∗ ( � C n ( D 2 m )) is a Z [ q ± 1 , t ± 1 ]-module. m , S 1 ), F = C n ( f ) lifts to � • For f ∈ Diff( D 2 C n ( D 2 m ) . • � F ∗ : H ∗ ( � m )) → H ∗ ( � C n ( D 2 C n ( D 2 m )) is Z [ q ± 1 , t ± 1 ]-linear . • This defines a representation of the braid group m , S 1 ) acting on H n ( � B m ≈ MCG ( D 2 C n ( D 2 m )) [Lawrence, Krammer ( n = 2), Bigelow].

Homological representations Surface braid groups Heisenberg groups Regular Torelli groups Configurations in the disc � � ( D 2 − { z 1 , . . . , z m } ) n − Diag • C n ( D 2 m ) = / S n , � C n ( D 2 m ) is a regular covering associated with a quotient map m ) , ∗ ) → Z 2 = < q , t > , ψ : π 1 ( C n ( D 2 defined using abelianisation of B n + m and B m . • H ∗ ( � C n ( D 2 m )) is a Z [ q ± 1 , t ± 1 ]-module. m , S 1 ), F = C n ( f ) lifts to � • For f ∈ Diff( D 2 C n ( D 2 m ) . • � F ∗ : H ∗ ( � m )) → H ∗ ( � C n ( D 2 C n ( D 2 m )) is Z [ q ± 1 , t ± 1 ]-linear . • This defines a representation of the braid group m , S 1 ) acting on H n ( � B m ≈ MCG ( D 2 C n ( D 2 m )) [Lawrence, Krammer ( n = 2), Bigelow]. • Variants: action on H n ( � m ) , H n ( � m )) , ∂ � C n ( D 2 C n ( D 2 C n ( D 2 m ))), H n ( � C n ( D 2 m )) , � ν ǫ ), . . .

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 = Σ g , 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 = Σ g , 1 on homology of the space of ordered configurations. • This reproduces the Johnson filtration. Let p 0 ∈ ∂ Σ, n > 0. Theorem (Moriyama) The kernel of the action of MCG on H n (Σ n , diag ∪ { contains p 0 } ) is the n-th Torelli subgroup T n (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 = Σ g , 1 on homology of the space of ordered configurations. • This reproduces the Johnson filtration. Let p 0 ∈ ∂ Σ, n > 0. Theorem (Moriyama) The kernel of the action of MCG on H n (Σ n , diag ∪ { contains p 0 } ) is the n-th Torelli subgroup T n (Σ) . • The kernel of the action on H n ((Σ n , diag ∪ { contains p 0 } ) / S n ) is the Torelli group T 1 (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 = Σ g , 1 on homology of the space of ordered configurations. • This reproduces the Johnson filtration. Let p 0 ∈ ∂ Σ, n > 0. Theorem (Moriyama) The kernel of the action of MCG on H n (Σ n , diag ∪ { contains p 0 } ) is the n-th Torelli subgroup T n (Σ) . • The kernel of the action on H n ((Σ n , diag ∪ { contains p 0 } ) / S n ) is the Torelli group T 1 (Σ). • Nice cell decomposition of the pair (Σ n , diag ∪ { contains p 0 } ) compatible with S n 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 = Σ g , 1 is an oriented genus g surface with one boundary component; C n (Σ) = (Σ n − Diag ) / S n , B n (Σ) = π 1 ( C n (Σ) , ∗ ).

Homological representations Surface braid groups Heisenberg groups Regular Torelli groups Surface braids • Σ = Σ g , 1 is an oriented genus g surface with one boundary component; C n (Σ) = (Σ n − Diag ) / S n , B n (Σ) = π 1 ( C n (Σ) , ∗ ). • Σ = D 2 / ∼ , where ∼ identifies edges according to the word c � g i =1 b i a i b i a i . A loop in C n (Σ) is represented by its graph in [0 , 1] × D 2 .

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: σ 1 , . . . , σ n − 1 , a 1 , . . . , a g , b 1 , . . . , b g . • Relations: usual braid relations on σ i and mixed relations below. ( R 1) a r σ i = σ i a r (1 ≤ r ≤ g ; i � = 1) ; b r σ i = σ i b r (1 ≤ r ≤ g ; i � = 1) ; ( R 2) σ 1 a r σ 1 a r = a r σ 1 a r σ 1 (1 ≤ r ≤ g ) ; σ 1 b r σ 1 b r = b r σ 1 b r σ 1 (1 ≤ r ≤ g ) ; σ − 1 1 a s σ 1 a r = a r σ − 1 ( R 3) 1 a s σ 1 ( s < r ) ; σ − 1 1 b s σ 1 b r = b r σ − 1 1 b s σ 1 ( s < r ) ; σ − 1 1 a s σ 1 b r = b r σ − 1 1 a s σ 1 ( s < r ) ; σ − 1 1 b s σ 1 a r = a r σ − 1 1 b s σ 1 ( s < r ) ; σ − 1 ( R 4) 1 a r σ 1 b r = b r σ 1 a r σ 1 (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 = H 1 (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 = H 1 (Σ) H = Z × H with ( k , h )( k ′ , h ′ ) = ( k + k ′ + h . h ′ , h + h ′ ). • �   1 X k • �   ⊂ SL g +2 ( Z ). H ≃ 0 I g Y 0 0 1