THE SIGNATURE MOD 2, 4 AND 8 Andrew Ranicki (Edinburgh) Larry - PDF document

THE SIGNATURE MOD 2, 4 AND 8 Andrew Ranicki (Edinburgh) Larry Taylor (Notre Dame) Oxford, 31st January 2005 1

The signature mod 2, 4 and 8 of a 4 k -dimensional Poincar´ e space X • Theorem σ ∗ ( X ) ≡ χ ( X ) (mod 2) with σ ∗ ( X ), χ ( X ) ∈ Z the signature and Euler characteristic. • Theorem σ ∗ ( X ) ≡ ⟨P 2 ( v ) , [ X ] ⟩ (mod 4) P 2 : H 2 k ( X ; Z 2 ) → H 4 k ( X ; Z 4 ) Pontrjagin square, v = v 2 k ( ν X ) ∈ H 2 k ( X ; Z 2 ) the 2 k th Wu class of the Spivak normal fibration ν X ⟨ x ∪ x, [ X ] ⟩ = ⟨ v ∪ x, [ X ] ⟩ ∈ Z 2 ( x ∈ H 2 k ( X ; Z 2 )) • Theorem σ ∗ ( X ) ≡ ⟨ � v ∪ � v, [ X ] ⟩ (mod 8) v ∈ H 2 k ( X ) of v . for any integral lift � • To what extent are these classical results for the signature of a Poincar´ e space true for the ‘mod 8 signature’ of a ‘normal space’? 2

Spherical fibrations • A spherical fibration is a Serre fibration ν : S j − 1 → E → X. • The Thom space T ( ν ) is the mapping cone of E → X . Will only consider oriented case, so have Thom class U ∈ � H j ( T ( ν )) with ∼ = � H ∗ ( X ) , U ∩ − : � H ∗ + j ( T ( ν )) ∼ = � � U ∪ − : H ∗ ( X ) H ∗ + j ( T ( ν )) . • Wu classes v r ( ν ) ∈ H r ( X ; Z 2 ) ( r � 0) characterized by dual Steenrod squares H r + j ( T ( ν ); Z 2 ) . χ ( Sq ) r ( U ) = U ∪ v r ( ν ) ∈ � • Spherical fibrations classified by maps ν : X → BSG ( j ). Stable classifying space → j BSG ( j ) , π ∗ ( BSG ) = π S BSG = lim ∗− 1 − with H ∗ ( BSG ), H ∗ ( BSG ) finite for ∗ ̸ = 0. 3

Normal spaces • Definition (Quinn, 1972) An n -dimensional normal space ( X, ν X , ρ X ) is a space X together with a spherical fibration ν X : X → BSG ( j ) and a map ρ X : S n + j → T ( ν X ). The fundamental class of X is the Hurewicz-Thom image H n + j ( T ( ν X )) ∼ [ X ] = U ∩ h ( ρ X ) ∈ � = H n ( X ) . • Thom-Wu formula: for any x ∈ H n − r ( X ; Z 2 ) [ X ] ∩ Sq r ( x ) = [ X ] ∩ ( v r ( ν X ) ∪ x ) ∈ H 0 ( X ; Z 2 ) • Will assume that the torsion-free quotients F r ( X ) = H r ( X ) / torsion are finitely gener- ated, e.g. if X is finite, or H r ( X ) is torsion. 4

Poincar´ e spaces • Definition An n -dimensional Poincar´ e space X is a finite CW complex with fundamen- tal class [ X ] ∈ H n ( X ) and duality isomor- phisms ∼ = � H ∗ ( X ) [ X ] ∩ − : H n −∗ ( X ) • Canonical example An oriented n -dimensional manifold is an n -dimensional Poincar´ e space. • Theorem (Spivak 1965, Wall, Browder) An n -dimensional Poincar´ e space X is an n -dimensional normal space, with ν X the ‘Spivak normal fibration’ ν X : S j − 1 → ∂W → W ≃ X defined by a regular neighbourhood ( W, ∂W ) of X ⊂ S n + j ( j large), and ρ X : S n + j → W/∂W ≃ T ( ν X ) the degree 1 collapse map. 5

Normal maps • A normal map of n -dimensional normal spaces ( f, b ) : X → Y is a degree 1 map f : X → Y f ∗ [ X ] = [ Y ] ∈ H n ( Y ) together with a map of normal fibrations b : ν X → ν Y s.t. T ( b ) ρ X = ρ Y ∈ π n + k ( T ( ν Y )) . • Proposition (Quinn) The mapping cylinder W of a n -dimensional normal map ( f, b ) : X → Y defines an ( n + 1)-dimensional normal space cobordism ( W ; X, Y ). • Basic question of surgery theory: is a Poincar´ e space homotopy equivalent to a manifold? Surgery obstruction to a normal map ( f, b ) : X → Y from a manifold X to a Poincar´ e space Y being bordant to a homotopy equiv- alence. Is a normal space bordant to a Poincar´ e space? Same obstruction. 6

The signature of a 4 k -dimensional normal space X • Symmetric intersection pairing ϕ : F 2 k ( X ) × F 2 k ( X ) → Z ; ( x, y ) �→ ⟨ x ∪ y, [ X ] ⟩ Nonsingular for Poincar´ e X . • The signature of X is σ ∗ ( X ) = signature( F 2 k ( X ) , ϕ ) ∈ Z • Warning For non-Poincar´ e X can have σ ∗ ( X ) ̸≡ χ ( X ) (mod 2) Proof For any finite CW complex X with odd χ ( X ) ∈ Z (e.g. X = {∗} ) and any ν X : X → BSG ( j ) set ρ X = ∗ : S 4 k + j → T ( ν X ), so that [ X ] = 0 ∈ H 4 k ( X ), σ ∗ ( X ) = 0 ∈ Z . 7

Normal and Poincar´ e cobordism (I) • Cobordism of normal and Poincar´ e spaces, with groups Ω N n , Ω P n . • Signature σ ∗ ( X ) ∈ Z is a Poincar´ e cobordism invariant, with mod 2 reduction χ ( X ) ∈ Z 2 • Theorem (Quinn) ‘Pontrjagin-Thom’ isomorphisms for normal space cobordism ∼ Ω N = � π n ( MSG ) ; ( X, ν X , ρ X ) �→ ν X ρ X n with MSG the Thom spectrum of the uni- versal spherical fibration 1 : BSG → BSG . Proof Every normal space ( X, ν X , ρ X ) is cobordant to ( BSG, 1 , ν X ρ X ) by mapping cylinder of normal map ν X : X → BSG . • The signature and mod 2 Euler character- istic are not normal space cobordism in- variants: F ∗ ( BSG )=0 ( ∗̸ =0), χ ( {∗} ) = 1. 8

Normal and Poincar´ e cobordism (II) • Theorem (Levitt-Jones-Quinn-Hausmann- Vogel, 1972-1988) For n � 4 there is an exact sequence · · · → L n ( Z ) → Ω P n → Ω N n → L n − 1 ( Z ) → . . . with L ∗ ( Z ) the simply-connected surgery obstruction groups. • Theorem (Brumfiel and Morgan, 1976) The signature and the mod-8-Hirzebruch number define surjections σ ∗ : Ω P 4 k → Z ; X �→ σ ∗ ( X ) , σ ∗ : Ω N 4 k → Z 8 ; X �→ ⟨ ν ∗ X ( ℓ 4 k ) , [ X ] ⟩ � with ℓ 4 k ∈ H 4 k ( BSG ; Z 8 ) the mod 8 ℓ -class. σ ∗ and � σ ∗ are isomorphisms for k = 1. The forgetful maps Ω P 4 k → Ω N 4 k ( k � 1) are surjections, since L 4 k − 1 ( Z ) = 0. 9

The mod 8 signature of a 4 k -dimensional normal space X • Definition The mod 8 signature is the Brumfiel- Morgan mod 8 Hirzebruch number σ ∗ ( X ) = ⟨ ν ∗ X ( ℓ 4 k ) , [ X ] ⟩ ∈ Z 8 . � • The mod 8 signature of a Poincar´ e X is the σ ∗ ( X ) = [ σ ∗ ( X )] ∈ Z 8 . signature mod 8, � • Every X is normal cobordant to a Poincar´ e σ ∗ ( X ) = [ σ ∗ ( Y )] ∈ Z 8 . space Y , with � • Warning For non-Poincar´ e X can have mod 8 signature ̸ = signature mod 8 σ ∗ ( X ) ̸ = [ σ ∗ ( X )] ∈ Z 8 . � Proof Take ν X = 1 : X = BSG ( j ) → BSG ( j ). σ ∗ ( X ) for Every d ̸ = 0 ∈ Z 8 is realized as d = � some ρ X : S 4 k + j → X , but σ ∗ ( X ) = 0 ∈ Z . 10

Homological formulae for the mod 2 and 4 signatures of normal spaces • Theorem 1 (R.-T.) The mod 4 reduction of the mod 8 signature of a 4 k -dimensional normal space X is σ ∗ ( X )] = ⟨P 2 ( v 2 k ( ν X )) , [ X ] ⟩ ∈ Z 4 [ � with P 2 : H 2 k ( X ; Z 2 ) → H 4 k ( X ; Z 4 ) the Pontrjagin square. (True for Poincar´ e X : Morita (1971), Brumfiel- Morgan (1974)) • Corollary (R.-T.) The mod 2 reduction of the mod 8 signature of a 4 k -dimensional normal space X is σ ∗ ( X )] = ⟨ v 2 k ( ν X ) ∪ v 2 k ( ν X ) , [ X ] ⟩ ∈ Z 2 [ � 11

Homological formulae for the mod 8 signature of certain normal spaces (I) Theorem 2 (R.-T.) Let X be a 4 k -dimensional normal space. Suppose that v 2 k ( ν X ) ∈ ker( δ 4 : H 2 k ( X ; Z 2 ) → H 2 k +1 ( X ; Z 2 )) = im( H 2 k ( X ; Z 4 ) → H 2 k ( X ; Z 2 )) , with δ 4 = the Bockstein for 0 → Z 2 → Z 4 → Z 2 → 0 For any lift v ∈ H 2 k ( X ; Z 4 ) of v 2 k ( ν X ) ∈ H 2 k ( X ; Z 2 ) σ ∗ ( X ) = ⟨P 4 ( v ) , [ X ] ⟩ ∈ Z 8 � with P 4 : H 2 k ( X ; Z 4 ) → H 4 k ( X ; Z 8 ) the Pontr- jagin square. 12

Homological formulae for the mod 8 signature of certain normal spaces (II) Corollary (R.-T.) Suppose that v 2 k ( ν X ) ∈ ker( δ ∞ : H 2 k ( X ; Z 2 ) → H 2 k +1 ( X )) = im( H 2 k ( X ) → H 2 k ( X ; Z 2 )) with δ ∞ = the Bockstein for 0 → Z 2 � Z → Z 2 → 0 . For any lift v ∈ H 2 k ( X ) of v 2 k ( ν X ) ∈ H 2 k ( X ; Z 2 ) ⟨ x ∪ x, [ X ] ⟩ = ⟨ v ∪ x, [ X ] ⟩ ∈ Z 2 ( x ∈ H 2 k ( X )) and σ ∗ ( X ) = ⟨ v ∪ v, [ X ] ⟩ ∈ Z 8 . � (True for Poincar´ e X : Hirzebruch and Hopf (1958), van der Blij (1959)) 13

Strategy of proofs (I) • Use the chain complex theory of algebraic surgery to interpret the mod 8 signature σ ∗ ( X ) ∈ Z 8 as the cobordism class of the � ‘algebraic normal complex’ ( C ( X ) , ϕ, γ, χ ) of X , computing it as a ‘characteristic num- ber’ of the ‘algebraic normal structure’ ( ϕ, γ, χ ). • ϕ = { ϕ s | s � 0 } consists of the chain map ϕ 0 = [ X ] ∩ − : C ( X ) 4 k −∗ → C ( X ) and the chain homotopies ϕ s +1 : ϕ s ≃ Tϕ s , which determine the evaluation of the Steenrod and Pontrjagin squares on the fundamental class [ X ] ∈ H 4 k ( X ). • γ is the ‘chain bundle’ of ν X : X → BSG ( j ), determined by Wu classes v ∗ ( ν X ) ∈ H ∗ ( X ; Z 2 ). χ is determined by ρ X ∈ π 4 k + j ( T ( ν X )). 14

Strategy of proofs (II) • ϕ and γ are essentially homological in nature, but χ is more subtle: difference be- tween ρ X ∈ π 4 k + j ( T ( ν X )) and the Hurewicz- Thom image U ∩ h ( ρ X ) = [ X ] ∈ H 4 k ( X ). • It turns out that the mod 4 reduction σ ∗ ( X )] ∈ Z 4 is determined by ϕ and γ , and [ � hence by P 2 : H 2 k ( X ; Z 2 ) → H 4 k ( X ; Z 4 ) as in Theorem 1. σ ∗ ( X ) ∈ Z 8 is • The mod 8 signature � in general determined by ϕ, γ and also χ . However, if the Bockstein hypothesis σ ∗ ( X ) is of Theorem 2 is satisfied then � determined only by ϕ, γ , and hence by P 4 : H 2 k ( X ; Z 4 ) → H 4 k ( X ; Z 8 ) as in the conclusion. 15