By Allan J. Sieradski
The remedy of the topic of this article isn't encyclopedic, nor was once it designed to be appropriate as a reference guide for specialists. really, it introduces the themes slowly of their historical demeanour, in order that scholars aren't beaten by means of the last word achievements of numerous generations of mathematicians. cautious readers will see how topologists have progressively subtle and prolonged the paintings in their predecessors and the way such a lot reliable rules achieve past what their originators predicted. To inspire the improvement of topological instinct, the textual content is abundantly illustrated. Examples, too various to be thoroughly lined in semesters of lectures, make this article appropriate for self sufficient examine and make allowance teachers the liberty to choose what they are going to emphasize. the 1st 8 chapters are compatible for a one-semester direction generally topology. the complete textual content is appropriate for a year-long undergraduate or graduate point curse, and offers a powerful origin for a next algebraic topology direction dedicated to the better homotopy teams, homology, and cohomology.
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Additional resources for An Introduction to Topology and Homotopy
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.
An Introduction to Topology and Homotopy by Allan J. Sieradski