Exotic spheres and the Kervaire invariant Addendum to the slides - PowerPoint PPT Presentation

1 Exotic spheres and the Kervaire invariant Addendum to the slides Michel Kervaire’s work in surgery and knot theory http://www.maths.ed.ac.uk/˜aar/slides/kervaire.pdf Andrew Ranicki (Edinburgh) 29th May, 2009

� � � � � � � � � � � � � � 2 The Kervaire-Milnor braid for m I. ◮ For any m � 5 there is a commutative braid of 4 interlocking exact sequences (slide 46) b 0 π m +1 ( G / PL )= P m +1 π m ( PL / O )=Θ m π m − 1 ( O ) � � � � � � � a � � � c � � � o � � � � � � � � � � � � � � � � � � � � � � � � � � � � � π m +1 ( G / O ) π m ( PL ) π m ( G / O )= A m � � � � � � � � � � � a � � o � � � � � � � � � � � � � � � � � � � � � � � � � � � � � π m ( G )= π S m =Ω fr π m ( O ) π m ( G / PL )= P m m J

3 The Kervaire-Milnor braid for m II. ◮ Θ m is the K-M group of oriented m -dimensional exotic spheres. ◮ P m = Z , 0 , Z 2 , 0 , Z , 0 , Z 2 , 0 , . . . is the m -dimensional simply-connected surgery obstruction group. These groups only depend on m (mod 4). ◮ a : A m = π m ( G / O ) → P m sends an m -dimensional almost framed differentiable manifold M to the surgery obstruction of the corresponding normal map ( f , b ) : M m → S m . ◮ For even m b : P m → Θ m − 1 sends a nonsingular ( − ) m / 2 -quadratic form over Z of rank r to the boundary Σ m − 1 = ∂ W of the Milnor plumbing W of r copies of τ S m / 2 realizing the form. ◮ The image of b is the subgroup bP m ⊆ Θ m − 1 of the ( m − 1)-dimensional exotic spheres Σ m − 1 which are the boundaries Σ m − 1 = ∂ W of m -dimensional framed differentiable manifolds W . ◮ c : Θ m → π m ( G / O ) sends an m -dimensional exotic sphere Σ m to its fibre-homotopy trivialized stable normal bundle.

4 The Kervaire-Milnor braid for m III. ◮ J : π m ( O ) → π m ( G ) = π S m is the J -homomorphism sending η : S m → O to the m -dimensional framed differentiable manifold ( S m , η ). ◮ The map o : π m ( G / O ) = A m → π m − 1 ( O ) sends an m -dimensional almost framed differentiable manifold M to the framing obstruction o ( M ) ∈ π m ( BO ) = π m − 1 ( O ) . ◮ The isomorphism π m ( PL / O ) → Θ m sends a vector bundle α : S m → BO ( k ) ( k large) with a PL trivialization β : α PL ≃ ∗ : S m → BPL ( k ) to the exotic sphere Σ m such that Σ m × R k is the smooth structure on the PL -manifold E ( α ) given by smoothing theory, with stable normal bundle α ν Σ m : Σ m ≃ S m � BO ( k ) ◮ π m ( PL ) = Θ fr m is the K-M group of framed n -dimensional exotic spheres.

� � � � � � � � � � � � � � � � � � 5 The Kervaire-Milnor braid for m = 4 k + 2 I. ◮ For m = 4 k + 2 � 5 the braid is given by b 0 J P 4 k +3 = 0 Θ 4 k +2 π 4 k +1 ( O ) π 4 k +1 ( G ) � � � � � � � � � c � � � � � � o � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � π 4 k +2 ( PL ) π 4 k +2 ( G / O ) π 4 k +1 ( PL ) � � � � � � � � � a � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � π 4 k +2 ( O ) = 0 π 4 k +2 ( G ) P 4 k +2 = Z 2 Θ 4 k +1 J K b with K the Kervaire invariant map.

6 The Kervaire-Milnor braid for m = 4 k + 2 II. ◮ K is the Kervaire invariant on the (4 k + 2)-dimensional stable homotopy group of spheres K : π 4 k +2 ( G ) = π S → j π j +4 k +2 ( S j ) 4 k +2 = lim − = Ω fr 4 k +2 = { framed cobordism } → P 4 k +2 = Z 2 ◮ K is the surgery obstruction: K = 0 if and only if every (4 k + 2)-dimensional framed differentiable manifold is framed cobordant to a framed exotic sphere. ◮ The exotic sphere group Θ 4 k +2 fits into the exact sequence � π 4 k +2 ( G ) K � Z 2 � Θ 4 k +2 � ker( π 4 k +1 ( PL ) → π 4 k +1 ( G )) � 0 0

7 The Kervaire-Milnor braid for m = 4 k + 2 III. ◮ a : π 4 k +2 ( G / O ) → Z 2 is the surgery obstruction map, sending a normal map ( f , b ) : M 4 k +2 → S 4 k +2 to the Kervaire invariant of M . ◮ b : P 4 k +2 = Z 2 → Θ 4 k +1 sends the generator 1 ∈ Z 2 to the boundary b (1) = Σ 4 k +1 = ∂ W of the Milnor plumbing W of two copies of τ S 2 k +1 � 1 � 1 using the standard rank 2 quadratic form over Z with Arf 0 1 invariant 1. ◮ The image of b is the subgroup bP 4 k +2 ⊆ Θ 4 k +1 of the (4 k + 1)-dimensional exotic spheres Σ 4 k +1 which are the boundaries Σ 4 k +1 = ∂ W of framed (4 k + 2)-dimensional differentiable manifolds W . If k is such that K = 0 (e.g. k = 2) then bP 4 k +2 = Z 2 ⊆ Θ 4 k +1 , and if Σ 4 k +1 = 1 ∈ bP 4 k +2 (as above) then M 4 k +2 = W ∪ Σ 4 k +1 D 4 k +2 is the (4 k + 2)-dimensional Kervaire PL manifold without a differentiable structure. ◮ c : Θ 4 k +2 → π 4 k +2 ( G / O ) sends a (4 k + 2)-dimensional exotic sphere Σ 4 k +2 to its fibre-homotopy trivialized stable normal bundle.

8 What if K = 0 ? ◮ For any k � 1 the following are equivalent: ◮ K : π 4 k +2 ( G ) = π S 4 k +2 → Z 2 is 0, ◮ Θ 4 k +2 ∼ = π 4 k +2 ( G ), ◮ ker( π 4 k +1 ( PL ) → π 4 k +1 ( G )) ∼ = Z 2 , ◮ Every simply-connected (4 k + 2)-dimensional Poincar´ e complex X with a vector bundle reduction ˜ ν X : X → BO of the Spivak normal fibration ν X : X → BG is homotopy equivalent to a closed (4 k + 2)-dimensional differentiable manifold. When is K � = 0 ? ◮ Theorem (Browder 1969) If K � = 0 then 4 k + 2 = 2 j − 2 for some j � 2. ◮ It is known that K � = 0 for 4 k + 2 ∈ { 2 , 6 , 14 , 30 , 62 } . ◮ Theorem (Hill-Hopkins-Ravenel 2009) If K � = 0 then 4 k + 2 ∈ { 2 , 6 , 14 , 30 , 62 , 126 } . ◮ It is not known if K = 0 or K � = 0 for 4 k + 2 = 126.

9 The exotic spheres home page http://www.maths.ed.ac.uk/˜aar/exotic.htm The Kervaire invariant home page http://www.math.rochester.edu/u/faculty/doug/kervaire.html