Math 527 - Homotopy Theory Hurewicz theorem
نویسنده
چکیده
Alternate proof. Using a bit of differential topology (or a more geometric construction along the lines of Hatcher § 4.1 Exercise 15), consider the degree of a smooth map f : S → S. Since every homotopy class [f ] contains a smooth representative, and all such maps have the same degree (i.e. degree is a homotopy invariant), this defines a function deg : πn(S )→ Z. One readily shows that deg is a group homomorphism. One can show moreover that two maps S → S with the same degree are homotopic, i.e. deg is injective. The equality deg([id]) = 1 shows that deg is surjective, hence an isomorphism.
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