Meyer’s function of the hyperelliptic mapping class group and related invariants of 3-manifolds Takayuki Morifuji Tokyo University of Agriculture and Technology CTQM, 28 March 2008 1
Secondary invariants signature cocycle Meyer’s function periodic ւ ↓ Z cov ց Torelli group η -invariant von Neumann ρ -inv Morita’s homomorphism signature op bdd coh class Casson inv CTQM, 28 March 2008 2
Contents • Signature cocycle and Meyer’s function • Eta-invariant • von Neumann rho-invariant • Casson invariant • Bounded cohomology CTQM, 28 March 2008 3
§ Signature cocycle Σ g : an oriented closed C ∞ -surface of genus g M g = π 0 Diff + Σ g mapping class group Fix a symplectic basis of H 1 (Σ g , Z ) r : M g → Sp(2 g, Z ) homology rep . I g = Ker r Torelli group ⋆ Meyer’s signature cocycle τ ∈ Z 2 (Sp(2 g, Z ) , Z ) CTQM, 28 March 2008 4
A, B ∈ Sp(2 g, Z ) , I : the identity matrix Define V A,B ⊂ R 2 g × R 2 g to be ( x, y ) | ( A − 1 − I ) x + ( B − I ) y = 0 � � V A,B = Define the pairing map on R 2 g × R 2 g by � ( x 1 , y 1 ) , ( x 2 , y 2 ) � A,B = ( x 1 + y 1 ) · J ( I − B ) y 2 , � O where · is the inner product in R 2 g , J = � I − I O ⇒ Symmetric bilinear form on V A,B CTQM, 28 March 2008 5
Define τ ( A, B ) = Sign ( V A,B , � , � A,B ) From Novikov additivity, τ ( A, B ) satisfies the cocycle condition, i.e. τ ( A, B ) + τ ( AB, C ) = τ ( A, BC ) + τ ( B, C ) ⇒ τ ∈ Z 2 (Sp(2 g, Z ) , Z ) signature cocycle CTQM, 28 March 2008 6
Properties of τ For A, B, C ∈ Sp(2 g, Z ) (i) ABC = I ⇒ τ ( A, B ) = τ ( B, C ) = τ ( C, A ) (ii) τ ( A, I ) = τ ( A, A − 1 ) = 0 (iii) τ ( B, A ) = τ ( A, B ) (iv) τ ( A − 1 , B − 1 ) = − τ ( A, B ) (v) τ ( CAC − 1 , CBC − 1 ) = τ ( A, B ) CTQM, 28 March 2008 7
Remark • We can regard τ as a 2-cocycle of M g by r W 4 • τ ( A, B ) = Sign ↓ Σ g , P is the pair of pants P 2 ∂P = M A ∪ M B ∪ − M AB mapping tori • By definition, τ is a bounded 2-cocycle (i.e. | τ | ≤ 2 g ) CTQM, 28 March 2008 8
Hyperelliptic mapping class group ι : hyperelliptic involution ∆ g = { f ∈ M g | fι = ιf } If g = 1 , 2 ⇒ ∆ g = M g CTQM, 28 March 2008 9
Fact H ∗ (∆ g , Q ) = 0 , ∗ = 1 , 2 Cohen, Kawazumi Hence [ τ ] has a finite order in H 2 (∆ g , Z ) Fact (2 g + 1) τ ∈ B 2 (∆ g , Z ) ⇒ there exists the uniquely defined mapping � � 1 m φ : ∆ g → 2 g + 1 Z = 2 g + 1 ∈ Q | m ∈ Z s.t. δφ = τ | ∆ g Meyer’s function of ∆ g CTQM, 28 March 2008 10
Remark • φ ( f − 1 ) = − φ ( f ) � 0 = φ ( ff − 1 ) = φ ( f ) + φ ( f − 1 ) − τ ( f, f − 1 ) � • φ is a class function of ∆ g i.e. φ ( hfh − 1 ) = φ ( f ) , f, h ∈ ∆ g φ ( hfh − 1 ) = φ ( h ) + φ ( fh − 1 ) − τ ( h, fh − 1 ) = φ ( h ) + φ ( f ) + φ ( h − 1 ) − τ ( f, h − 1 ) − τ ( h, fh − 1 ) = φ ( f ) CTQM, 28 March 2008 11
⇒ an invariant of surface bundles over the circle • δφ = τ | ∆ g implies φ is a homomorphism on the Torelli group I g ∩ ∆ g ( g ≥ 2) � For � f, h ∈ I g ∩ ∆ g φ ( fh ) = φ ( f ) + φ ( h ) − τ ( f, h ) = φ ( f ) + φ ( h ) CTQM, 28 March 2008 12
a presentation of ∆ g Birman-Hilden Example generator : ζ i (1 ≤ i ≤ 2 g + 1) relation : ζ i ζ i +1 ζ i = ζ i +1 ζ i ζ i +1 ζ i ζ j = ζ j ζ i ( | i − j | ≥ 2) ( ζ 1 · · · ζ 2 g +1 ) 2 g +2 = 1 2 g +1 · · · ζ 1 ) 2 = 1 ( ζ 1 · · · ζ 2 ζ i commutes with ζ 1 · · · ζ 2 2 g +1 · · · ζ 1 CTQM, 28 March 2008 13
⋆ ζ i conjugate each other (in fact ζ i +1 = ξζ i ξ − 1 for ξ = ζ 1 · · · ζ 2 g +1 ) φ ( ζ i ) = g + 1 (for any i ) 2 g + 1 CTQM, 28 March 2008 14
Put φ ( ζ ) = φ ( ζ i ) . Using a defining relation of ∆ g 2 · · · ζ 1 ) 0 = φ ( ζ 1 · · · ζ 2 g +1 = φ ( ζ 1 · · · ζ 2 g +1 ) + φ ( ζ 2 g +1 · · · ζ 1 ) − τ ( ζ 1 · · · ζ 2 g +1 , ζ 2 g +1 · · · ζ 1 ) = 2 { (2 g + 1) φ ( ζ ) − 1 } − 2 g = 2(2 g + 1) φ ( ζ ) − 2( g + 1) φ ( ζ ) = g + 1 = ⇒ 2 g + 1 CTQM, 28 March 2008 15
⋆ ψ h = ( ζ 1 · · · ζ 2 h ) 4 h +2 ∈ ∆ g Dehn twist along a bounding simple closed curve 4 φ ( ψ h ) = − 2 g + 1 h ( g − h ) BSCC map CTQM, 28 March 2008 16
✓ ✏ ∆ 1 ∼ g = 1 = M 1 = SL (2 , Z ) ✒ ✑ ⋆ Meyer, Kirby-Melvin, Sczech · · · explicit formula of τ and φ φ ( A ) = − 1 3Ψ( A ) + σ ( A ) · 1 2(1 + sgn(tr A )) Ψ : SL (2 , Z ) → Z the Rademacher function � − 2 c � � a b � a − d σ ( A ) = Sign for A = a − d 2 b c d CTQM, 28 March 2008 17
⋆ Atiyah · · · geometric meanings of φ “The logarithm of the Dedekind η -function” Math. Ann. 278 (1987), 335–380 Various inv. ass. to SL (2 , Z ) coincide with φ cf . Rademacher ′ s function , Hirzebruch signature defect , the special value of Shimizu L funct . η invariant & its adiabatic limit , etc . CTQM, 28 March 2008 18
✓ ✏ g ≥ 2 Geometric aspects of φ ? ✒ ✑ ⋆ periodic auto. (of finite order) ⇒ η -invariant (mapping torus) ⋆ Z -covering ⇒ von Neumann ρ -invariant (1 st MMM class & Rochlin inv) ⋆ Torelli group ⇒ Casson invariant (Heegaard splitting) CTQM, 28 March 2008 19
Related works ⋆ Kasagawa, Iida · · · other construction of φ for g = 2 ⋆ Matsumoto, Endo · · · the loc. sign. of hyp. Lefschetz fibrations ⋆ Kuno, Sato · · · Meyer’s function in other settings CTQM, 28 March 2008 20
§ Eta-invariant M : ori. closed Riem 3-mfd → η ( M ) is defined Thm Atiyah-Patodi-Singer W : a cpt ori Riem 4-mfd s.t. ∂W = M , product near M η ( M ) = 1 � p 1 − Sign W 3 W p 1 : 1 st Pontrjagin form of the metric CTQM, 28 March 2008 21
Remark If W is closed ⇒ Sign W = 1 � W p 1 3 For f ∈ M g M f = Σ g × R / ( x, t ) ∼ ( f ( x ) , t +1) mapping torus Thm f ∈ ∆ g periodic ⇒ η ( M f ) = φ ( f ) Σ g × S 1 ↓ finite Riem cov M f CTQM, 28 March 2008 22
This is shown by using the following formula. f ∈ M g : periodic auto. of the order n n − 1 η ( M f ) = 1 � f, f k � � τ n k =1 Moreover if f ∈ ∆ g n − 1 � 0 = φ ( id ) = φ ( f n ) = nφ ( f ) − f, f k � � τ k =1 CTQM, 28 March 2008 23
1 Cor f ∈ M g periodic, f ∈ ∆ g ⇒ η ( M f ) ∈ 2 g + 1 Z Example there exists f ∈ M 3 of order 3 s.t. its quotient orbifold ≈ S 2 (3 , 3 , 3 , 3 , 3) Then direct computation shows η ( M f ) = − 2 ∈ 1 3 / 7 Z ⇒ f / ∈ ∆ 3 CTQM, 28 March 2008 24
§ Relation to von Neumann rho-invariant Γ : a discrete group M : an ori closed Riem 3-mfd π 1 M → Γ : a surjective homo ⇒ Γ → ˆ M → M : Γ -covering → η (2) ( ˆ − M ) is defined von Neumann or L 2 η -invariant CTQM, 28 March 2008 25
Def & Thm Cheeger-Gromov η (2) ( ˆ M ) − η ( M ) is independent of a Riem metric || ρ (2) ( ˆ M ) von Neumann rho-invariant Remark ρ (2) ( ˆ M ) is an extension of rho-invariant η γ : the η -invariant ass. to γ : π 1 M → U ( n ) ⇒ ρ = η γ − nη is independent of a Riem metric CTQM, 28 March 2008 26
For f ∈ ∆ g Z → ˆ M f → M f Z -covering associated to π 1 M f → π 1 S 1 ⋆ φ is not multiplicative for coverings φ ( f k ) − kφ ( f ) Thm ρ (2) ( ˆ M f ) = lim k k →∞ Using the thm stated before and the approximation thm of the η -inv, due to Vaillant, L¨ uck-Schick CTQM, 28 March 2008 27
Γ ⊲ Γ 1 ⊲ Γ 2 ⊲ · · · : descending sequence s.t. [Γ : Γ k ] < ∞ and ∩ k Γ k = { 1 } M ( k ) = ˆ M/ Γ k → M : Γ / Γ k -covering Thm Vaillant, L¨ uck-Schick � � η M ( k ) η (2) ( ˆ M ) = lim [Γ : Γ k ] k →∞ CTQM, 28 March 2008 28
Example g = 1 A ∈ SL (2 , Z ) (1) Elliptic case ( | tr A | < 2 ) Let A n ∈ SL (2 , Z ) have the order n � − 1 � 0 � 0 � � � − 1 − 1 − 1 A 3 = , A 4 = , A 6 = 1 0 1 0 1 1 2 / 3 n = 3 ρ (2) ( ˆ M A n ) = 1 n = 4 4 / 3 n = 6 CTQM, 28 March 2008 29
� 1 � b (2) Parabolic case ( | tr A | = 2 ) A b = ( b ∈ Z ) 0 1 � − b/ | b | b � = 0 ρ (2) ( ˆ M A b ) = − sgn( b ) = 0 b = 0 (3) Hyperbolic case ( | tr A | > 2 ) ρ (2) ( ˆ M A ) = 0 ( φ ( A k ) = kφ ( A ) holds) CTQM, 28 March 2008 30
Cor If f ∈ I g ∩ ∆ g ⇒ ρ (2) ( ˆ M f ) = 0 ( φ is a homomorphism on I g ∩ ∆ g ) Remark If we restrict the above thm to the level 2 subgroup, we can obtain a relation among von Neumann rho-inv, 1 st MMM class and Rochlin inv in a framework of the bdd cohomology � � b ( S 1 , Z ) ∼ f ∗ e 1 “ = ” µ ( M f ) − ρ (2) ( ˆ M f ) in H 2 = R / Z f ∈ M g (2) = ker {M g → Sp(2 g, Z / 2) } CTQM, 28 March 2008 31
§ Casson invariant g ≥ 2 λ : { M | ori homology 3-sphere } → Z λ ( M ) ∼ # { π 1 M → SU (2) irr rep } /conj ⋆ Theory of characteristic classes of surface bundles we can consider the Casson inv of Z HS 3 from the view point of M g Morita K g = � BSCC map � ⊂ I g bounding simple closed curve CTQM, 28 March 2008 32
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