Construct a surjective map \(S^n \to S^n\) of degree zero, for each \(n \geq 1\).
Solution
Choose an embedded equatorial \((n-1)\)-sphere in \(S^n\) and collapse it to a point. This gives the pinch map \[ p:S^n\longrightarrow S^n\vee S^n. \] Let \[ g:S^n\vee S^n\longrightarrow S^n \] be the identity on the first summand and a reflection on the second summand. Set \[ f=g\circ p. \]
1. The map \(f\) is surjective.
The first wedge summand is mapped by \(g\) via the identity onto all of \(S^n\). Hence the composite \(f\) is onto.
2. The degree of \(f\) is zero.
On top homology, the pinch map sends a fundamental class to the sum of the fundamental classes of the two wedge summands: \[ p_*[S^n]=([S^n],[S^n]). \] The identity has degree \(+1\), while a reflection has degree \(-1\). Therefore \[ f_*[S^n]=(1+(-1))[S^n]=0, \] so \[ \boxed{\deg f=0.} \]
Thus for every \(n\ge1\) there is a surjective degree-zero map \(S^n\to S^n\).