Show that every map f: S^n→ S^n is homotopic to a multiple of the identity map by the following steps.(a) Use Lemma 4.10 (or simplicial approximation, Theorem 2 C .1 ) to reduce to the case that there exists a point q ∈ S^n with f^-1(q)={p1, ⋯, pk} and f is an invertible linear map near each pi(b) For f as in (a), consider the composition g f where g: S^n→ S^n collapses the complement of a small ball about q to the basepoint. Use this to reduce (a) further to the case k=1.(c) Finish the argument
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