Characteristic classes of homological surface bundles and four-dimensional topology Shigeyuki MORITA based on jw/w Takuya SAKASAI and Masaaki SUZUKI October 25, 2016 Shigeyuki MORITA Characteristic classes of homological surface bundles
Contents Contents An “enlargement” H g, 1 of the mapping class group 1 Representations of H g, 1 2 Kontsevich’s theorem and homology of Out F n 3 Characteristic classes of homological surface bundles 4 Prospect 5 Shigeyuki MORITA Characteristic classes of homological surface bundles
An “enlargement” H g, 1 of the mapping class group (1) Mapping class groups: M g = π 0 Diff + Σ g , M g, 1 = π 0 Diff(Σ g , D 2 ) Another description: rel ∂ ∼ M g, 1 = { (Σ g, 1 × I, ϕ ) ; ϕ : Σ g, 1 = Σ g, 1 × { 1 }} / isotopy Shigeyuki MORITA Characteristic classes of homological surface bundles
An “enlargement” H g, 1 of the mapping class group (1) Mapping class groups: M g = π 0 Diff + Σ g , M g, 1 = π 0 Diff(Σ g , D 2 ) Another description: rel ∂ ∼ M g, 1 = { (Σ g, 1 × I, ϕ ) ; ϕ : Σ g, 1 = Σ g, 1 × { 1 }} / isotopy Group of homology cobordism classes of homology cylinders: Garoufalidis-Levine (based on Goussarov and Habiro): rel ∂ ∼ H g, 1 = { ( homology Σ g, 1 × I, ϕ ) ; ϕ : Σ g, 1 = Σ g, 1 × { 1 }} / homology cobordism Shigeyuki MORITA Characteristic classes of homological surface bundles
An “enlargement” H g, 1 of the mapping class group (2) two versions: big kernel ， Freedman H top H smooth − − − − − − − − − − − − − → g, 1 g, 1 surjective enlargements of M g, 1 Shigeyuki MORITA Characteristic classes of homological surface bundles
An “enlargement” H g, 1 of the mapping class group (2) two versions: big kernel ， Freedman H top H smooth − − − − − − − − − − − − − → g, 1 g, 1 surjective enlargements of M g, 1 Θ 3 := { homology 3 -spheres } / smooth homology cobordism infinite rank by Furuta, Fintushel-Stern Define a group H g, 1 by the following central extension 0 → Θ 3 = H smooth → H smooth → H g, 1 → 1 0 , 1 g, 1 Shigeyuki MORITA Characteristic classes of homological surface bundles
An “enlargement” H g, 1 of the mapping class group (3) Problem Study the Euler class χ ( H smooth ) ∈ H 2 ( H g, 1 ; Θ 3 ) g, 1 Shigeyuki MORITA Characteristic classes of homological surface bundles
An “enlargement” H g, 1 of the mapping class group (3) Problem Study the Euler class χ ( H smooth ) ∈ H 2 ( H g, 1 ; Θ 3 ) g, 1 One of the foundational results of Freedman: Theorem (Freedman) Any homology 3 -sphere bounds a contractible topological 4 -manifold so that Θ 3 top = 0 → H top It follows that H smooth g, 1 factors through H g, 1 g, 1 Shigeyuki MORITA Characteristic classes of homological surface bundles
An “enlargement” H g, 1 of the mapping class group (4) Θ 3 → H smooth Freedman → H top → H g, 1 − − − − − − g, 1 g, 1 Problem (about “Picard groups” ) Study the following homomorphisms ( g ≥ 3 ) Harer H 2 ( H top g, 1 ) → H 2 ( H g, 1 ) → H 2 ( H smooth ) → H 2 ( M g, 1 ) ∼ = Z g, 1 Shigeyuki MORITA Characteristic classes of homological surface bundles
An “enlargement” H g, 1 of the mapping class group (4) Θ 3 → H smooth Freedman → H top → H g, 1 − − − − − − g, 1 g, 1 Problem (about “Picard groups” ) Study the following homomorphisms ( g ≥ 3 ) Harer H 2 ( H top g, 1 ) → H 2 ( H g, 1 ) → H 2 ( H smooth ) → H 2 ( M g, 1 ) ∼ = Z g, 1 ∼ ∞ -rank? ∞ -rank? = ? Shigeyuki MORITA Characteristic classes of homological surface bundles
Representations of H g, 1 (1) Theorem (Dehn-Nielsen-Zieschang) • M g ∼ = Out + π 1 Σ g (outer automorphism group) • M g, 1 ∼ = { ϕ ∈ Aut π 1 Σ g, 1 ; ϕ ( ζ ) = ζ } ζ : boundary curve “differentiate” ⇒ Shigeyuki MORITA Characteristic classes of homological surface bundles
Representations of H g, 1 (1) Theorem (Dehn-Nielsen-Zieschang) • M g ∼ = Out + π 1 Σ g (outer automorphism group) • M g, 1 ∼ = { ϕ ∈ Aut π 1 Σ g, 1 ; ϕ ( ζ ) = ζ } ζ : boundary curve “differentiate” ⇒ ✓ ✏ Definition (“Lie algebra” of M g, 1 ) h g, 1 = { symplectic derivation of the free Lie algebra L ( H Q ) } ✒ ✑ ∞ ⊕ h g, 1 = h g, 1 ( k ) : symplectic derivation Lie algebra of L ( H Q ) k =0 very important in low dimensional topology Shigeyuki MORITA Characteristic classes of homological surface bundles
Representations of H g, 1 (2) Mal’cev nilpotent completion of π 1 Σ g, 1 : · · · → N d +1 → N d → · · · → N 1 = H Q → 0 ( H Q = H 1 (Σ g, 1 ; Q )) ⇒ obtain a series of representations of M g, 1 : ρ ∞ = { ρ d } d : M g, 1 → lim Aut 0 N d ( ρ d : M g, 1 → Aut 0 N d ) ← − d →∞ Shigeyuki MORITA Characteristic classes of homological surface bundles
Representations of H g, 1 (2) Mal’cev nilpotent completion of π 1 Σ g, 1 : · · · → N d +1 → N d → · · · → N 1 = H Q → 0 ( H Q = H 1 (Σ g, 1 ; Q )) ⇒ obtain a series of representations of M g, 1 : ρ ∞ = { ρ d } d : M g, 1 → lim Aut 0 N d ( ρ d : M g, 1 → Aut 0 N d ) ← − d →∞ associated embedding of Lie algebras: ✓ ✏ ∞ ⊕ small ideal h + τ : M g, 1 ( d ) / M g, 1 ( d + 1) ⊂ ⊂ h g, 1 g, 1 d =1 ✒ ✑ M g, 1 ( d ) := Ker ρ d Johnson filtration Shigeyuki MORITA Characteristic classes of homological surface bundles
Representations of H g, 1 (3) Stallings’ theorem ⇒ Theorem (Garoufalidis-Levine, Habegger) There exists a homomorphism ρ ∞ : H top ˜ g, 1 → lim Aut 0 N d ← − d →∞ ρ d : H top which extends ρ ∞ , each finite factor ˜ g, 1 → Aut 0 N d is surjective over Z for any d ≥ 1 Shigeyuki MORITA Characteristic classes of homological surface bundles
Representations of H g, 1 (3) Stallings’ theorem ⇒ Theorem (Garoufalidis-Levine, Habegger) There exists a homomorphism ρ ∞ : H top ˜ g, 1 → lim Aut 0 N d ← − d →∞ ρ d : H top which extends ρ ∞ , each finite factor ˜ g, 1 → Aut 0 N d is surjective over Z for any d ≥ 1 τ d M g, 1 ( d ) − − − − − − − → h g, 1 ( d ) image small � � � � ∩ τ d ˜ H top g, 1 ( d ) − − − − − → h g, 1 ( d ) surjective Shigeyuki MORITA Characteristic classes of homological surface bundles
Representations of H g, 1 (4) ρ ∞ M g, 1 − − − − − → lim Aut 0 N d ← − injective d →∞ � � � � ∩ surjective ρ ∞ ˜ → H top H smooth − − − − − − − − − → lim Aut 0 N d g, 1 g, 1 ← − d →∞ ⇒ obtain Shigeyuki MORITA Characteristic classes of homological surface bundles
Representations of H g, 1 (4) ρ ∞ M g, 1 − − − − − → lim Aut 0 N d ← − injective d →∞ � � � � ∩ surjective ρ ∞ ˜ → H top H smooth − − − − − − − − − → lim Aut 0 N d g, 1 g, 1 ← − d →∞ ⇒ obtain d →∞ H ∗ (Aut 0 N d ) → H ∗ ( H top ρ ∗ ˜ ∞ : lim g, 1 ; Q ) d →∞ H ∗ (Aut 0 N d ) → H ∗ ( H top ρ ∗ ˜ ∞ : lim g →∞ lim g, 1 ; Q ) Shigeyuki MORITA Characteristic classes of homological surface bundles
Representations of H g, 1 (5) Aut 0 N d is a linear algebraic group and we have Aut 0 N d ∼ = IAut 0 N d ⋊ Sp(2 g, Q ) Lie(IAut 0 N d ) ∼ = h + g, 1 [ d ] ( truncated ) Proposition ∞ , 1 ) Sp ⊗ H ∗ (Sp(2 ∞ , Q ); Q ) d →∞ H ∗ (Aut 0 N d ) ∼ c (ˆ = H ∗ h + g →∞ lim lim � h + h + g →∞ h + ∞ , 1 : completion of ∞ , 1 = lim g, 1 Shigeyuki MORITA Characteristic classes of homological surface bundles
Representations of H g, 1 (5) Aut 0 N d is a linear algebraic group and we have Aut 0 N d ∼ = IAut 0 N d ⋊ Sp(2 g, Q ) Lie(IAut 0 N d ) ∼ = h + g, 1 [ d ] ( truncated ) Proposition ∞ , 1 ) Sp ⊗ H ∗ (Sp(2 ∞ , Q ); Q ) d →∞ H ∗ (Aut 0 N d ) ∼ c (ˆ = H ∗ h + g →∞ lim lim � h + h + g →∞ h + ∞ , 1 : completion of ∞ , 1 = lim g, 1 ⇒ obtain ∞ , 1 ) Sp ⊗ H ∗ (Sp(2 ∞ , Q ); Q ) → H ∗ ( H top c (ˆ h + ρ ∗ ∞ : H ∗ ˜ g, 1 ; Q ) Shigeyuki MORITA Characteristic classes of homological surface bundles
Kontsevich’s theorem and homology of Out F n (1) ✓ ✏ Lie version of Kontsevich graph homology ✒ ✑ By using theory of Outer Space due to Culler and Vogtmann: Theorem (Kontsevich, Lie version) ∞ , 1 ) Sp 2 n ∼ PH k c ( � h + = H 2 n − k (Out F n +1 ; Q ) ⇒ ⊕ ∞ , 1 ) Sp ∼ H ∗ c ( � h + = Λ H ∗ (Out F n ; Q ) n ≥ 2 Λ : free associative algebra degree ( x ) = 2 n − 2 − k ( x ∈ H k (Out F n ; Q )) Shigeyuki MORITA Characteristic classes of homological surface bundles
Kontsevich’s theorem and homology of Out F n (2) ⊕ h ∞ , 1 ) dual H 2 n − 3 (Out F n ; Q ) ⇔ PH 1 ⇔ H 1 ( h + c ( � ∞ , 1 ) Sp n ≥ 2 Culler-Vogtmann: vcd(Out F n ) = 2 n − 3 Problem What are the generators: H 1 ( h + ∞ , 1 ) for the Lie algebra h + ∞ , 1 ? Shigeyuki MORITA Characteristic classes of homological surface bundles
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