By Nicolas Bourbaki
Bourbaki Library of Congress Catalog #66-25377 published in France 1966
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This quantity grew from a dialogue by way of the editors at the trouble of discovering strong thesis difficulties for graduate scholars in topology. even though at any given time we each one had our personal favourite difficulties, we said the necessity to supply scholars a much wider choice from which to settle on an issue unusual to their pursuits.
This textbook in aspect set topology is geared toward an upper-undergraduate viewers. Its mild velocity might be worthy to scholars who're nonetheless studying to jot down proofs. necessities contain calculus and a minimum of one semester of research, the place the scholar has been appropriately uncovered to the guidelines of simple set idea equivalent to subsets, unions, intersections, and services, in addition to convergence and different topological notions within the genuine line.
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Additional resources for Elements of mathematics. General topology. Part 1
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.
Elements of mathematics. General topology. Part 1 by Nicolas Bourbaki