Show that every map f:Sn→Sn is homotopic to a multiple of the identity map by the following steps.
((a)) Use Lemma 4.10 (or simplicial approximation, Theorem 2C.1) to reduce to the case that there exists a point q∈Sn 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 gf where g:Sn→Sn 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 by showing that an invertible n×n matrix can be joined by a path of such matrices to either the identity matrix or the matrix of a reflection. (Use Gaussian elimination, for example.)