By Tangora M.C. (ed.)
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Extra info for Algebraic Topology: Oaxtepec 1991
5. The set p(S n ) ⊂ Πn is a subgroup of the stable homotopy group Πn . For any homotopy sphere Σ the set p(Σ) is a coset of this subgroup p(S n ). Thus the correspondence Σ → p(Σ) deﬁnes a homomorphism p from Θn to the quotient group Πn /p(S n ). Proof. 4 with the identities (1) S n #S n = S n , (2) S n #Σ = Σ, (3) Σ#(−Σ) ∼ S n , we obtain p(S n ) + p(S n ) ⊂ p(S n ), (1) which shows that p(S n ) is a subgroup of Πn ; p(S n ) + p(Σ) ⊂ p(Σ), (2) which shows that p(Σ) is a union of cosets of this subgroup; and p(Σ) + p(−Σ) ⊂ p(S n ), (3) which shows that p(Σ) must be a single coset.
Kervaire and J. Milnor ϕ more carefully, taking particular care not to lose s-parallelizability in the process. Before starting the proof, it is convenient to sharpen the concepts of s-parallelizable manifold, and of spherical modiﬁcation. Definition. A framed manifold (M, f ) will mean a diﬀerentiable manifold M together with a ﬁxed trivialization f of the stable tangent bundle τM ⊕ εM . Now consider a spherical modiﬁcation χ(ϕ) of M . Recall that M and M = χ(M, ϕ) together bound a manifold W = (M × [0, 1]) ∪ (Dp+1 × Dq+1 ), where the subset S p × Dq+1 of Dp+1 × Dq+1 is pasted onto M × 1 by the imbedding ϕ (compare Milnor ).
2. Now let M be a 2k-manifold which is (k − 1)-connected. Then ψ : H k (M, bM ) → H 2k (M, bM ; π2k−1 (S k )) π2k−1 (S k ) is deﬁned. 3. Let k be odd1 and let M be s-parallelizable. Then an imbedded k-sphere in M has trivial normal bundle if and only if its dual cohomology class v ∈ H k (M, bM ) satisﬁes the condition ψ(v) = 0. 1 This lemma is actually true for even k also. August 26, 2009 16:21 44 9in x 6in b789-ch02 M. Kervaire and J. Milnor Proof. Let N be a closed tubular neighborhood of the imbedded sphere, and let M0 = M − Interior N.
Algebraic Topology: Oaxtepec 1991 by Tangora M.C. (ed.)