QUADRATIC STRUCTURES IN SURGERY THEORY Andrew Ranicki (Edinburgh) - PowerPoint PPT Presentation

1 QUADRATIC STRUCTURES IN SURGERY THEORY Andrew Ranicki (Edinburgh) ICMS, 5th July, 2006 ◮ 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)

2 Past ◮ The chain complex theory developed in The algebraic theory of surgery (R., 1980) expressed the surgery obstruction groups L ∗ ( A ) as the cobordism groups of ‘quadratic Poincar´ e complexes’, chain complexes C with quadratic Poincar´ e duality ψ . ◮ The Wall 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 σ ∗ ( f , b ) ∈ L m ( Z [ π 1 ( X )]) was expressed as the cobordism class of a quadratic Poincar´ e complex ( C , ψ ) obtained directly from ( f , b ), without preliminary surgeries below the middle dimension. The homology of C consists of the kernel Z [ π 1 ( X )]-modules H ∗ ( C ) = K ∗ ( M ) = ker( � f ∗ : H ∗ ( � M ) → H ∗ ( � X )) .

3 Advantages and a disadvantage ◮ The algebraic theory of surgery did indeed offer the advantages predicted by Wall, such as all kinds of exact sequences. ◮ However, the identification σ ∗ ( f , b ) = ( C , ψ ) was not as nice as could have been wished for! ◮ Specifically, 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 Z 2 -equivariant homotopy theory.

4 Present ◮ The ‘geometric Hopf invariant’ h ( F ) of Michael Crabb (Aberdeen) provides a satisfactory homotopy-theoretic foundation for algebraic surgery theory. ◮ Let X , Y be pointed spaces. The geometric Hopf invariant of a stable map F : Σ ∞ X → Σ ∞ Y is a stable map h ( F ) : Σ ∞ X → Σ ∞ (( S ∞ ) + ∧ Z 2 ( Y ∧ Y )) with good naturality properties: if π is a group, X , Y are π -spaces and F is π -equivariant then h ( F ) is π -equivariant. ◮ The quadratic structure of a normal map ( f , b ) : M → X is the evaluation ψ = ( h ( F ) /π )[ X ] with π = π 1 ( X ), X + → Σ ∞ � F : Σ ∞ � M + a stable π -equivariant map inducing the Umkehr f ! : C ( � X ) → C ( � M ) and h ( F ) /π : H m ( X ) → H m ( S ∞ × Z 2 ( � M × π � M )) . The resulting quadratic Poincar´ e complex ( C , ψ ) has a direct connection with double points of immersions g : S n � M m .

5 The Umkehr chain map ◮ The Umkehr of a map f : N → M of geometric Poincar´ e complexes is the ‘wrong-way’ Z [ π 1 ( M )]-module chain map � f ∗ � C ( � N ) m −∗ ≃ C ( � f ! : C ( � M ) ≃ C ( � M ) m −∗ N ) ∗− m + n N = f ∗ � with � M the universal cover of M , � M the pullback cover of N , m = dim M , n = dim N and M ) ∗ = Hom Z [ π 1 ( M )] ( C ( � C ( � M ) , Z [ π 1 ( M )]) . ◮ In the cases of interest f ! is induced by a stable map F , and the geometric Hopf invariant h ( F ) captures the double point class of an immersion, and the quadratic structure of a normal map.

6 The stable Umkehr of an immersion ◮ An immersion f : N n � M m has a normal bundle ν f : N → BO ( m − n ) with f ∗ τ M = τ N ⊕ ν f . ◮ For some k � 0 (e.g. if k � 2 n − m + 1) can approximate f → R k × M , with e : N → R k and by an embedding ( e , f ) : N ֒ ν ( e , f ) = ν f ⊕ ǫ k : N → BO ( m − n + k ) . ◮ Let � M be the universal cover of M . The Pontrjagin-Thom construction applied to the π 1 ( M )-equivariant embedding → R k × � N = f ∗ � e , � f ) : � ( � M ֒ M is a π 1 ( M )-equivariant stable Umkehr map to the Thom space M + → T ( ν ( ❡ F : Σ k � f ) ) = Σ k T ( ν ❡ f ) e , ❡ inducing f ! C (Σ k � F : ˙ M + ) ≃ C ( � � ˙ f ) )) ≃ C ( � M ) ∗− k C ( T ( ν ( ❡ N ) ∗− m + n − k . e , ❡ ◮ If f is an embedding can take k = 0, and F is unstable.

7 The stable Umkehr of a normal map ◮ The algebraic mapping cone C = C ( f ! ) of the Umkehr f ! : C ( � X ) → C ( � M ) of a degree 1 map f : M → X of m -dimensional geometric Poincar´ e complexes is such that H ∗ ( C ) = K ∗ ( M ) = ker( � f ∗ : H ∗ ( � M ) → H ∗ ( � X )) with � f ∗ a surjection split by f ! . ◮ For a manifold M and a normal map ( f , b ) : M → X f ! is induced by a π 1 ( X )-equivariant S -dual F : Σ k � X + → Σ k � M + of the map T ( � b ) : T ( ν ❡ M ) → T ( ν ❡ X ) of Thom spaces. ◮ F can also be constructed geometrically: apply Wall’s π - π theorem to obtain a homotopy equivalence ( X × D k , X × S k − 1 ) ≃ ( W , ∂ W ) ( k � 3) with ( W , ∂ W ) an ( m + k )-dimensional manifold with boundary. For k � 2 n − m + 1 approximate ( f , b ) by a framed embedding M ֒ → W and apply the Pontrjagin-Thom construction to � → � M ֒ W .

8 The quadratic construction on a space ◮ The quadratic construction on a space X is Q ( X ) = S ∞ × Z 2 ( X × X ) with the generator T ∈ Z 2 acting by T : S ∞ = lim S k → S ∞ ; s �→ − s , − → k T : X × X → X × X ; ( x , y ) �→ ( y , x ) . ◮ Let X + = X ⊔ { + } , i.e. X with an adjoined base point +. ◮ The reduced quadratic construction on a pointed space Y is Q ( Y ) = ( S ∞ ) + ∧ Z 2 ( Y ∧ Y ) . ˙ In particular Q ( X + ) = Q ( X ) + . ˙

9 Unstable vs. stable homotopy theory ◮ Given pointed spaces X , Y let [ X , Y ] be the set of homotopy classes of maps X → Y . ◮ The stable homotopy group is [Σ k X , Σ k Y ] = [ X , Ω ∞ Σ ∞ X ] { X ; Y } = lim − → k ◮ The stabilization map [ X , Y ] → { X ; Y } = [ X , Ω ∞ Σ ∞ Y ] is in general not an isomorphism! ◮ The quotient of Y ֒ → Ω ∞ Σ ∞ Y has a filtration, much studied by homotopy theorists. If f : N n � M m is an immersion with Umkehr stable map F : Σ ∞ M + → Σ ∞ T ( ν f ), the adjoint adj( F ) : M + → Ω ∞ Σ ∞ T ( ν f ) sends the k -tuple points of M to the k -th filtration.

10 The James-Hopf map � ∞ ◮ (1950’s) James decomposition ΩΣ Y ≃ s ( Y ∧ · · · ∧ Y ). k =1 ◮ (1970’s) Snaith and others constructed a stable homotopy equivalence � ∞ E Σ + Ω ∞ Σ ∞ Y ≃ s k ∧ Σ k ( Y ∧ · · · ∧ Y ) k =1 for connected Y , group completion in general. ◮ The stable homotopy projection Σ ∞ Ω ∞ Σ ∞ Y → Σ ∞ ( E Σ + 2 ∧ Σ 2 ( Y ∧ Y )) ( E Σ 2 = S ∞ ) is the James-Hopf double point map. However, only defined for connected Y , and not natural in Y . ◮ In order to get the quadratic structure of a normal map ( f , b ) : M → X need to split off the quadratic part of the X + → Ω ∞ Σ ∞ � π 1 ( X )-equivariant map adj( F ) : � M + .

11 The geometric Hopf invariant h ( F ) ◮ Let X , Y be pointed π -spaces. When is a k -stable π -map F : Σ k X → Σ k Y homotopic to the k -fold suspension Σ k F 0 of an unstable π -map F 0 : X → Y ? ◮ The geometric Hopf invariant of F is the stable π × Z 2 -equivariant map h ( F ) = ( F ∧ F )∆ X − ∆ Y F : Σ k , k X → Σ k , k ( Y ∧ Y ) 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 ) . ◮ The stable Z 2 -equivariant homotopy class of h ( F ) /π : Σ k , k X /π → Σ k , k ( Y ∧ π Y ) is the primary obstruction to the k-fold desuspension of F.

12 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 ◮ Example The Z 2 -equivariant Pontrjagin-Thom isomorphism identifies { S 0 ; S 0 } Z 2 with 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 an isomorphism � � | D Z 2 | , | D | − | D Z 2 | { S 0 ; S 0 } Z 2 ∼ = Z ⊕ Z ; D = D Z 2 ∪ ( D − D Z 2 ) �→ 2

13 Z 2 -equivariant stable homotopy theory = fixed-point + fixed-point-free ◮ Theorem (Crabb+R.) For any pointed π -spaces X , Y there is a naturally split short exact sequence of abelian groups ρ � { X /π ; Y /π } � { X /π ; ˙ � { X /π ; Y ∧ π Y } Z 2 � 0 0 Q ( Y ) /π } ◮ The surjection ρ is given by the Z 2 -fixed points, and is split by { X /π ; Y /π } → { X /π ; Y ∧ π Y } Z 2 ; F �→ ∆ Y F . ◮ The injection is induced by the projection S ∞ → {∗} { X /π ; ˙ Q ( Y ) /π } = { X /π ; ( S ∞ ) + ∧ Y ∧ π Y } Z 2 → { X /π ; Y ∧ π Y } Z 2 .