Show that every map \(f: S^n \to S^n\) is homotopic to a multiple of the identity map by the following steps.
Use Lemma 4.10 (or simplicial approximation, Theorem 2C.1) to reduce to the case that there exists a point \(q \in S^n\) with \(f^{-1}(q) = \{p_1, \ldots, p_k\}\) and \(f\) is an invertible linear map near each \(p_i\).
For \(f\) as in (a), consider the composition \(gf\) where \(g: S^n \to S^n\) collapses the complement of a small ball about \(q\) to the basepoint. Use this to reduce (a) further to the case \(k = 1\).
Finish the argument by showing that an invertible \(n \times n\) matrix can be joined by a path of such matrices to either the identity matrix or the matrix of a reflection.
Solution
Let \(f:S^n\to S^n\).
(a) Reduction to finitely many regular preimages
By Lemma 4.10 (equivalently, after simplicial approximation and a small local modification), \(f\) is homotopic to a map for which there is a point \(q\in S^n\) such that \[ f^{-1}(q)=\{p_1,\dots,p_k\}, \] and, in coordinate balls around each \(p_i\) and around \(q\), the map is given by an invertible linear map. Choose pairwise disjoint small balls \(B_i\) about the \(p_i\)’s and a ball \(B\) about \(q\) so that \[ f|_{B_i}:B_i\to B \] is linear and invertible in these coordinates.
(b) Split into one-preimage pieces
Let \[ g:S^n\to S^n \] collapse the complement of the interior of \(B\) to the basepoint. This quotient map has degree \(1\), hence is homotopic to the identity. Thus \[ gf\simeq f. \]
The map \(gf\) is constant off the union of the \(B_i\)’s. Collapsing the complement of the interiors of the \(B_i\)’s identifies the domain with a wedge \[ S^n\vee\cdots\vee S^n \] of \(k\) spheres, and \(gf\) factors as \[ S^n\xrightarrow{\text{pinch}}\bigvee_{i=1}^k S^n \xrightarrow{h_1\vee\cdots\vee h_k}S^n, \] where each \(h_i\) has exactly one preimage of \(q\) and is linear near that point. In \(\pi_n(S^n)\), this says \[ [f]=[h_1]+\cdots+[h_k]. \] So it remains to classify the case \(k=1\).
(c) The one-preimage case
For such a map, the local model near the unique preimage is an invertible matrix \[ A\in GL_n(\mathbb R). \] The group \(GL_n(\mathbb R)\) has exactly two path-components, distinguished by the sign of the determinant. Gaussian elimination through invertible matrices (or polar decomposition) gives a path in \(GL_n(\mathbb R)\) from \(A\) to \[ I\quad\text{if }\det A>0, \] and to a reflection \[ R=\operatorname{diag}(-1,1,\dots,1) \quad\text{if }\det A<0. \] Using this path inside the local ball and keeping the map constant outside gives a homotopy of \(h_i\) to the identity map when \(\det A>0\), and to a reflection when \(\det A<0\). The latter represents \(-[\operatorname{id}_{S^n}]\) in \(\pi_n(S^n)\).
Hence, if \[ \varepsilon_i=\operatorname{sign}(\det Df_{p_i})\in\{\pm1\}, \] then \[ [f]=\sum_{i=1}^k\varepsilon_i[\operatorname{id}_{S^n}]. \] Writing \[ d=\sum_i\varepsilon_i\in\mathbb Z, \] we obtain \[ \boxed{f\simeq d\cdot\operatorname{id}_{S^n}.} \] Thus every self-map of \(S^n\) is homotopic to an integral multiple of the identity.