THE GEOMETRIC HOPF INVARIANT Michael Crabb (Aberdeen) Andrew - PowerPoint PPT Presentation

1 THE GEOMETRIC HOPF INVARIANT Michael Crabb (Aberdeen) Andrew Ranicki (Edinburgh) G¨ ottingen, 29th September, 2007

2 The algebraic theory of surgery ◮ “The chain complex theory offers many advantages . . . a simple and satisfactory algebraic version of the whole setup. I hope it can be made to work.” C.T.C. Wall, Surgery on Compact Manifolds (1970) ◮ The chain complex theory developed in The algebraic theory of surgery (R., 1980) expressed surgery obstruction of a normal map ( f , b ) : M → X from an m -dimensional manifold M to an m -dimensional geometric Poincar´ e complex X as the cobordism class of a quadratic Poincar´ e complex ( C , ψ ) σ ∗ ( f , b ) = ( C , ψ ) ∈ L m ( Z [ π 1 ( X )]) with C a f.g. free Z [ π 1 ( X )]-module chain complex such that H ∗ ( C ) = K ∗ ( M ) = ker( � f ∗ : H ∗ ( � M ) → H ∗ ( � X )) and ψ : H ∗ ( C ) ∼ = H m −∗ ( C ) an algebraic Poincar´ e duality. ◮ Originally, it was necessary to make ( f , b ) highly-connected by preliminary surgeries below the middle dimension.

3 Advantages and a disadvantage ◮ The algebraic theory of surgery did indeed offer the advantages predicted by Wall in 1970. ◮ However, the identification σ ∗ ( f , b ) = ( C , ψ ) was not as nice as could have been wished for! ◮ The chain homotopy theoretic treatment of the Wall self-intersection function counting double points Z [ π 1 ( M )] µ ( g : S n � M 2 n ) ∈ { x − ( − ) n x − 1 | x ∈ π 1 ( M ) } was too indirect, making use of Wall’s result that for n � 3 µ ( g ) = 0 if and only if g is regular homotopic to an embedding – proved by the Whitney trick for removing double points. ◮ Need to count double points of immersions using π 1 ( M ) × Z 2 -equivariant homotopy theory, specifically an equivariant version of the geometric Hopf invariant.

4 Unstable vs. stable homotopy theory ◮ The stabilization map [Σ k X , Σ k Y ] = [ X , Ω ∞ Σ ∞ Y ] [ X , Y ] → { X ; Y } = lim − → k is in general not an isomorphism! ◮ Terminology: for any space X let X + = X ⊔ { + } (disjoint union) and X ∞ = X ∪ {∞} (one point compactification). ◮ Ω ∞ Σ ∞ Y / Y is filtered, with k th filtration quotient F k ( Y ) = E Σ + k ∧ Σ k ( ∧ k Y ) . ◮ The Thom space of a j -plane bundle R j → E ( ν ) → M is T ( ν ) = E ( ν ) ∞ , and F k ( T ( ν )) = T ( e k ( ν )) with � � R jk → E ( e k ( ν )) = E Σ k × Σ k E ( ν ) → E Σ k × Σ k M . k k ◮ For an immersion f : M m � N n with ν f : M → BO ( n − m ) and Umkehr map F : Σ ∞ N + → Σ ∞ T ( ν f ) the adjoint N + → Ω ∞ Σ ∞ T ( ν f ) sends k -tuple points of f to F k ( T ( ν f )).

5 The Hopf invariant (I.) ◮ (Hopf, 1931) Isomorphism H : π 3 ( S 2 ) ∼ = Z via linking numbers of S 1 ⊔ S 1 ֒ → S 3 . ◮ (Freudenthal, 1937) Suspension map for pointed space X E : π n ( X ) → π n +1 (Σ X ) ; ( f : S n → X ) �→ (Σ f : S n +1 → Σ X ) . angung ). If X is ( m − 1)-connected then E is an ( E for Einh¨ isomorphism for n � 2 m − 2 and surjective for n = 2 m − 1. ◮ (G.W.Whitehead, 1950) EHP exact sequence � π n ( X ) E � π n +1 (Σ X ) H � π n ( X ∧ X ) P � π n − 1 ( X ) � . . . . . . for any ( m − 1)-connected space X , with n � 3 m − 2. ◮ For X = S m , n = 2 m H = Hopf invariant : π 2 m +1 ( S m +1 ) → π 2 m ( S m ∧ S m ) = Z .

6 The quadratic construction ◮ Given an inner product space V let LV = V with Z 2 -action T : LV → LV ; v �→ − v with restriction T : S ( LV ) → S ( LV ). ◮ The quadratic construction on pointed space X is Q V ( X ) = S ( LV ) + ∧ Z 2 ( X ∧ X ) with T : X ∧ X → X ∧ X ; ( x , y ) �→ ( y , x ). The projection Q V ( X ) = S ( LV ) + ∧ ( X ∧ X ) → Q V ( X ) is a double cover away from the base point. ◮ Q R 0 ( X ) = { pt. } , Q R 1 ( X ) = X ∧ X . ◮ Q R k ( S 0 ) = S ( L R k ) + / Z 2 = ( RP k − 1 ) + . ◮ For V = R ∞ write Q ( X ) = Q R ∞ ( X ) = lim Q R k ( X ) . − → k

7 The Hopf invariant (II.) ◮ (James, 1955) Stable homotopy equivalence for connected X � ∞ ΩΣ X ≃ s ( X ∧ · · · ∧ X ) . k =1 ◮ (Snaith, 1974) Stable homotopy equivalence � ∞ E Σ + Ω ∞ Σ ∞ X ≃ s k ∧ Σ k ( X ∧ · · · ∧ X ) . k =1 for connected X . Group completion for disconnected X . ◮ For k = 2 a stable homotopy projection Ω ∞ Σ ∞ X → Q ( X ) = E Σ + 2 ∧ Σ 2 ( X ∧ X ) . However, until now it was only defined for connected X , and was not natural in X .

8 The stable Z 2 -equivariant homotopy groups ◮ Given pointed Z 2 -spaces X , Y let [ X , Y ] Z 2 be the set of Z 2 -equivariant homotopy classes of Z 2 -equivariant maps X → Y . ◮ The stable Z 2 -equivariant homotopy group is [Σ k , k X , Σ k , k Y ] Z 2 { X ; Y } Z 2 = lim − → k with T : Σ k , k X = S k ∧ S k ∧ X → Σ k , k X ; ( s , t , x ) �→ ( t , s , x ) , T : Σ k , k ( Y ∧ Y ) → Σ k , k ( Y ∧ Y ) ; ( s , t , y 1 , y 2 ) �→ ( t , s , y 2 , y 1 ) . ◮ Example By the Z 2 -equivariant Pontrjagin-Thom isomorphism { S 0 ; S 0 } Z 2 = the cobordism group of 0-dimensional framed Z 2 -manifolds (= finite Z 2 -sets). The decomposition of finite Z 2 -sets as fixed ∪ free determines � � | D Z 2 | , | D | − | D Z 2 | { S 0 ; S 0 } Z 2 ∼ = Z ⊕ Z ; D = D Z 2 ∪ ( D − D Z 2 ) �→ 2

9 The relative difference ◮ For any inner product space V there is a cofibration S 0 = { 0 } + → V ∞ → V ∞ / { 0 } + = Σ S ( V ) + with S ( V ) = { v ∈ V | � v � = 1 } and tu ∼ � V ∞ / { 0 } + ; ( t , u ) �→ [ t , u ] = = Σ S ( V ) + 1 − t . ◮ For maps p , q : V ∞ ∧ X → Y such that p (0 , x ) = q (0 , x ) ∈ Y ( x ∈ X ) define the relative difference map δ ( p , q ) : Σ S ( V ) + ∧ X → Y ; � p ([1 − 2 t , u ] , x ) if 0 � t � 1 / 2 ( t , u , x ) �→ q ([2 t − 1 , u ] , x ) if 1 / 2 � t � 1 . ◮ The homotopy class of δ ( p , q ) is the obstruction to the existence of a rel 0 ∞ ∧ X homotopy p ≃ q : V ∞ ∧ X → Y . Barratt-Puppe sequence · · · → [Σ S ( V ) + ∧ X , Y ] → [ V ∞ ∧ X , Y ] → [ X , Y ]

10 Z 2 -equivariant stable homotopy theory = fixed-point + fixed-point-free ◮ Theorem For any pointed spaces X , Y there is a split short exact sequence of abelian groups 1+ T � { X ; Y ∧ Y } Z 2 ρ � { X ; Y } → 0 0 → { X ; Q ( Y ) } with an S -duality isomorphism { X ; Q ( Y ) } ∼ [Σ S ( LV ) + ∧ V ∞ ∧ X , V ∞ ∧ LV ∞ ∧ ( Y ∧ Y )] Z 2 . lim = − → V , dim( V ) < ∞ ◮ ρ is given by the Z 2 -fixed points, split by σ : { X ; Y } → { X ; Y ∧ Y } Z 2 ; F �→ ∆ Y F . ◮ The injection 1 + T is induced by projection S ( L R ∞ ) + → 0 ∞ 1 + T : { X ; Q ( Y ) } = { X ; Q ( Y ) } Z 2 → { X ; Y ∧ Y } Z 2 split by δ : { X ; Y ∧ Y } Z 2 → { X ; Q ( Y ) } ; G �→ δ ( G , σρ ( G )) .

11 The geometric Hopf invariant h ( F ) (I.) ◮ Let X , Y be pointed spaces. The geometric Hopf invariant of a stable map F : Σ ∞ X → Σ ∞ Y is the stable map h ( F ) = δ (( F ∧ F )∆ X , ∆ Y F ) : Σ ∞ X → Σ ∞ Q ( Y ) . ◮ The injection 1 + T : { X ; Q ( Y ) } ֒ → { X ; Y ∧ Y } Z 2 sends the stable homotopy class of h ( F ) to the stable Z 2 -equivariant homotopy class of (1 + T ) h ( F ) = ∆ Y F − ( F ∧ F )∆ X : X → Y ∧ Y . ◮ The stable homotopy class of h ( F ) is the primary obstruction to the desuspension of F. ◮ Good naturality properties: if π is a group, X , Y are π -spaces and F is π -equivariant then h ( F ) is π -equivariant.

12 The geometric Hopf invariant h ( F ) (II.) ◮ Proposition The geometric Hopf invariant of F : Σ ∞ X → Σ ∞ Y h ( F ) ∈ ker( ρ : { X ; Y ∧ Y } Z 2 → { X ; Y } ) = im(1 + T : { X ; Q ( Y ) } ֒ → { X ; Y ∧ Y } Z 2 ) has the following properties: (i) If F ∈ im([ X , Y ] → { X ; Y } ) then h ( F ) = 0. (ii) For F 1 , F 2 : Σ ∞ X → Σ ∞ Y h ( F 1 + F 2 ) = h ( F 1 ) + h ( F 2 ) + ( F 1 ∧ F 2 )∆ X . (iii) For F : Σ ∞ X → Σ ∞ Y , G : Σ ∞ Y → Σ ∞ Z h ( GF ) = ( G ∧ G ) h ( F ) + h ( G ) F . (iv) If X = S 2 m , Y = S m , F : S 2 m + ∞ → S m + ∞ then h ( F ) = mod 2 Hopf invariant ( F ) ∈ { S 2 m ; Q ( S m ) } = Z 2 . (v) h : { X ; Y } → { X ; Q ( Y ) } ; F �→ h ( F ) is the James-Hopf double point map.

13 The Main Theorem ◮ Theorem The quadratic Poincar´ e complex ( C , ψ ) of an m -dimensional normal map ( f , b ) : M → X has ψ = ( e ⊗ e )( h ( F ) /π )[ X ] ∈ Q m ( C ) = H m ( C ( S ( L R ∞ )) ⊗ Z [ Z 2 ] ( C ⊗ Z [ π ] C )) with π = π 1 ( X ), [ X ] ∈ H m ( X ) the fundamental class, and h ( F ) /π : H m ( X ) → H m ( S ( L R ∞ ) × Z 2 ( � M × π � M )) the π -equivariant geometric Hopf invariant. Here X + → Σ ∞ � M + is the stable π -equivariant map F : Σ ∞ � inducing the Umkehr f ! : C ( � X ) → C ( � M ) determined by b : ν M → ν X , and e = inclusion : C ( � M ) → C = C ( f ! ). ◮ The m -dimensional quadratic Poincar´ e complex ( C , ψ ) has a direct connection with double points of immersions S n � M m , particularly for m = 2 n .