THE MOD 8 SIGNATURE OF A SURFACE BUNDLE Andrew Ranicki Report on a - PowerPoint PPT Presentation

1 THE MOD 8 SIGNATURE OF A SURFACE BUNDLE Andrew Ranicki Report on a joint project with Dave Benson, Caterina Campagnolo and Carmen Rovi http://www.maths.ed.ac.uk/˜aar/ University of Edinburgh Dedicated to the memory of Fritz Hirzebruch on his 89th birthday Bonn, 17th October, 2016

2 Introduction ◮ The signature of an oriented m -dimensional manifold with boundary ( M , ∂ M ) is { signature ( H m / 2 ( M ) , φ ) if m ≡ 0(mod4) σ ( M ) = 0 otherwise , φ : H m / 2 ( M ) × H m / 2 ( M ) → Z symmetric intersection form. ◮ The non-multiplicativity of σ for a fibre bundle F → E → B σ ( E ) − σ ( B ) σ ( F ) ∈ Z has been studied for 60 years: Chern, Hirzebruch and Serre (1956), Kodaira (1969), Atiyah (1970), Hirzebruch (1970), Meyer (1972), Hambleton, Korzeniewski and R. (2005) . . . ◮ Particularly interesting for a surface bundle g S 1 × S 1 → E → B = Σ h F = Σ g = # with σ (Σ g ) = 0 by definition. In general, σ ( E ) ̸ = 0 ∈ Z .

3 The Meyer signature class ◮ In his 1972 Bonn thesis Werner Meyer (a student of Hirzebruch) constructed the signature class τ ∈ H 2 (Sp(2 g , Z ); Z ) . ◮ The signature of a surface bundle Σ g → E → Σ h is the evaluation σ ( E ) = ⟨ f ∗ τ, [Σ h ] ⟩ ∈ Z with f : π 1 (Σ h ) → Sp(2 g , Z ) = Aut Z ( H 1 (Σ g ) , φ ) the monodromy action, and φ : H 1 (Σ g ) × H 1 (Σ g ) → Z ; ( x , y ) �→ ⟨ x ∪ y , [Σ g ] ⟩ . the nonsingular symplectic intersection form over Z .

4 Divisibility by 4, but not by 8 in general ◮ Meyer also constructed an explicit cocycle for the signature class τ , and computed  if g = 1  Z 12  τ = 4 ∈ H 2 (Sp(2 g , Z ); Z ) = Z ⊕ Z 2 if g = 2   Z if g � 3. ◮ The signature of Σ g → E → Σ h is divisible by 4 σ ( E ) ∈ 4 Z ⊂ Z ◮ Every multiple of 4 arises as σ ( E ) for some E . ◮ The image of τ/ 4 in H 2 (Sp(2 g , Z ); Z 2 ) = Z 2 ( g � 4) determines the mod 8 signature σ ( E ) = ⟨ f ∗ τ, [Σ h ] ⟩ ∈ 4 Z / 8 Z = Z 2 . ◮ Carmen Rovi (Edinburgh Ph.D. thesis, 2015) identified σ ( E ) / 4 ∈ Z 2 with an Arf-Kervaire invariant.

5 The mod 8 signature and group cohomology ◮ Problem Does there exist a class τ k ∈ H 2 (Sp(2 g , Z k ); Z 8 ) for the mod 8 signature for some k � 2, such that k [ τ ] = 4 ∈ H 2 (Sp(2 g , Z k ); Z 8 ) = Z 8 ? τ k = p ∗ with p k = projection : Z → Z k . Posed for k = 2 by Klaus and Teichner. ◮ If there exists such a class τ k then the mod 8 signature σ ( E ) = ⟨ f ∗ k τ k , [Σ h ] ⟩ ∈ 4 Z 8 ⊂ Z 8 depends only on the mod k monodromy action f k : π 1 (Σ h ) → Sp(2 g , Z ) → Sp(2 g , Z k ) . ◮ k = 2 will not do , since H 2 (Sp(2 g , Z 2 ); Z 8 ) = 0 ( g � 4).

6 The mod 8 signature class ◮ Theorem 1 (BCRR, 2016) k = 4 will do. The mod 8 signature class τ 4 = 4 ∈ H 2 (Sp(2 g , Z 4 ); Z 8 ) = Z 8 is such that σ ( E ) = ⟨ f ∗ 4 τ 4 , [Σ h ] ⟩ ∈ 4 Z 8 ⊂ Z 8 f � Sp(2 g , Z ) � Sp(2 g , Z 4 ) . with f 4 : π 1 (Σ h ) ◮ Proof It is enough to show that � H 2 (Sp(2 g , Z ); Z 8 ) = Z 8 , τ ∈ H 2 (Sp(2 g , Z ); Z ) = Z ∼ = � H 2 (Sp(2 g , Z ); Z 8 ) = Z 8 τ 4 ∈ H 2 (Sp(2 g , Z 4 ); Z 8 ) = Z 8 have the same images. ◮ Easy , but no cocycle and no geometry!

7 The mapping torus T ( α ) ◮ The mapping class group of Σ g is defined as usual by Mod g = π 0 (Homeo + (Σ g )) with Homeo + (Σ g ) the group of orientation-preserving homeomorphisms α : Σ g → Σ g . ◮ The mapping torus of α ∈ Mod g is the closed oriented 3-manifold T ( α ) = Σ g × I / { ( x , 0) ∼ ( α ( x ) , 1) | x ∈ Σ g } Total space of fibre bundle Σ g → T ( α ) → S 1 .

8 The double mapping torus T ( α, β ) ◮ The double mapping torus T ( α, β ) of α, β ∈ Mod g is the total space of the fibre bundle Σ g → T ( α, β ) → P = pair of pants , an oriented 4-manifold with boundary ∂ T ( α, β ) = T ( α ) ⊔ T ( β ) ⊔ − T ( αβ ) T(β) T(α, β) T(αβ) T(α)

9 A cocycle for τ ∈ H 2 ( Sp (2 g , Z ); Z ) ◮ Theorem (Meyer, 1972) The Wall non-additivity of the signature formula gives σ ( T ( α, β )) = σ (ker((1 − α − 1 1 − β ) : H ⊕ H → H ) , Φ) H = H 1 (Σ g ) , Φ(( x 1 , y 1 ) , ( x 2 , y 2 )) = φ ( x 1 + y 1 , (1 − β )( y 2 )) . ◮ The function τ : Sp(2 g , Z ) × Sp(2 g , Z ) → Z ; ( α, β ) �→ σ ( T ( α, β )) is a cocycle for the signature class τ ∈ H 2 (Sp(2 g , Z ); Z ).

10 The idea of proof of Meyer’s Theorem ◮ For a surface bundle Σ g → E → Σ h with monodromy π 1 (Σ h ) = ⟨ α 1 , β 1 , . . . , α h , β h | [ α 1 , β 1 ] . . . [ α h , β h ] ⟩ → Mod g lift the decomposition ∑ = h 2 D 2 D P P P 1 2 2h 4 h ∪ E = D 2 × Σ g ∪ ω i − 1 , ω i ) ∪ D 2 × Σ g to T ( � (simplified) i =1 with � ω i the i th factor in [ α 1 , β 1 ] . . . [ α h , β h ] and ω i the product of the first i factors. ∑ 4 h ◮ By Novikov additivity σ ( E ) = − σ ( T ( � ω i − 1 , ω i )) ∈ Z . i =1

11 The Brown-Kervaire invariant BK ( V , b , q ) ∈ Z 8 ◮ Defined by E.H.Brown (1972) for a nonsingular symmetric form ( V , b ) over Z 2 with Z 4 -valued quadratic refinement q (f.g. free Z 2 -module V , b : V × V → Z 2 , q : V → Z 4 ) by the Gauss sum ∑ √ dim Z 2 V e 2 π iBK ( V , b , q ) / 8 ∈ C e 2 π iq ( x ) / 4 = 2 x ∈ V ◮ The mod 8 signature of a nonsingular symmetric form ( H , φ ) over Z is σ ( H , φ ) = BK ( H / 2 H , b , q ) ∈ Z 8 with b ( x , y ) = [ φ ( x , y )] , q ( x ) = [ φ ( x , x )] .

12 A cocycle for τ 4 ∈ H 2 (Sp(2 g , Z 4 ); Z 2 ) ◮ The verification that Meyer’s function τ : Sp(2 g , Z ) × Sp(2 g , Z ) → Z is a cocycle used the Novikov additivity for the signature of the union of manifolds with boundary σ ( M ∪ ∂ M = − ∂ M ′ M ′ ) = σ ( M ) + σ ( M ′ ) ∈ Z . ◮ Our cocycle τ 4 : Sp(2 g , Z 4 ) × Sp(2 g , Z 4 ) → Z 2 is constructed using the Z 8 -valued Brown-Kervaire invariant, for which there is no analogue of Novikov additivity .

13 Mapping tori are boundaries ◮ Ω 3 = 0: every closed oriented 3-dimensional manifold is the boundary of an oriented 4-manifold, so there exists a function δ T : Mod g → { oriented 4-manifolds with boundary } ; α �→ δ T ( α ) such that ∂δ T ( α ) = T ( α ). ◮ So for any α, β ∈ Mod g have closed oriented 4-dimensional manifold T ( α, β ) ∪ ( δ T ( α ) ⊔ δ T ( β ) ⊔ δ T ( αβ )) T(β) δT(β) T(α, β) T(αβ) δT(αβ) δT(α) T(α)

14 The mod 8 signature cocycle ◮ Theorem 2 (BCRR, 2016) For any δ T the function Mod g × Mod g → Z 8 ; ( α, β ) �→ BK ( T ( α, β ) ∪ δ T ( α ) ∪ δ T ( β ) ∪ − δ T ( αβ )) is a cocycle for the pullback of 4 [ τ ] ∈ H 2 (Sp(2 g , Z 4 ); Z 8 ) 4 τ 4 = p ∗ along the Z 4 -coefficient monodromy Mod g → Sp(2 g , Z 4 ). ◮ Very implicit, since it relies on the choice of bounding 4-manifolds δ T ( α ). In general, not divisible by 4. ◮ Algebraic Poincar´ e cobordism to the rescue .

15 Algebraic Poincar´ e cobordism ◮ (R., 1980- . . . ) For any ring with involution A { L n ( A ) L n ( A ) = cobordism groups of n -dimensional f.g. free A -module { symmetric chain equivalence C n −∗ → C chain complexes with a quadratic ◮ 1 + T : L n ( A ) = Wall surgery obstruction group → L n ( A ). ◮ L 0 ( A ) (resp. L 0 ( A )) = Witt group of nonsingular symmetric (resp. quadratic) forms over A ◮ For A = Z signature σ : L 0 ( Z ) ∼ = Z with 1 + T = 8 : L 0 ( Z ) = Z → L 0 ( Z ) = Z . ◮ For A = Z 4 Brown-Kervaire invariant BK : L 0 ( Z 4 ) ∼ = Z 8 with 1 + T = 4 : L 0 ( Z 4 ) = Z 2 → L 0 ( Z 4 ) = Z 8 . ◮ Symmetric signature Ω n → L n ( Z ) → L n ( Z 4 ).

16 Generalized signature cocycle and class via algebra ◮ Manifolds with boundary, union, mapping torus and double mapping torus all have analogues in the world of algebraic Poincar´ e cobordism, for any ring A . ◮ The algebraic mapping torus gives morphism T : Sp(2 g , A ) → L 3 ( A ) ; α �→ T ( α ) . ◮ Theorem 3 (BCRR, 2016) If L 3 ( A ) = 0 the algebraic double mapping torus gives a class τ A ∈ H 2 (Sp(2 g , A ); L 4 ( A )) with cocycle τ A : Sp(2 g , A ) × Sp(2 g , A ) → L 4 ( A ) ; ( α, β ) �→ τ A ( α, β ) = T ( α, β ) ∪ δ T ( α ) ∪ δ T ( β ) ∪ − δ T ( αβ ) for any choice of α �→ δ T ( α ) with ∂δ T ( α ) = T ( α ).

17 e cobordism of A = Z The algebraic Poincar´ ◮ L 3 ( Z ) = 0. Canonical null-cobordism δ T ( α ) for algebraic T ( α ) with Euler characteristic χ ( α ) = dim Z ker(1 − α : Z 2 g → Z 2 g ) ( α ∈ Sp(2 g , Z )) . ◮ Isomorphism σ : L 4 ( Z ) → Z ; ( C , φ ) �→ σ ( H 2 ( C ) , φ 0 ) . ◮ (Turaev 1985) The cocycle τ : Sp(2 g , Z ) × Sp(2 g , Z ) → Z ; ( α, β ) �→ στ Z ( α, β ) − ( χ ( α ) + χ ( β ) − χ ( αβ )) is divisible by 4, representing the Meyer signature class τ = 4 ∈ H 2 (Sp(2 g , Z ); Z ) = Z .