Degree zero does not imply a sphere map is constant

Choose two disjoint closed round $n$-balls $D_{+},D_{-}$ in $S^n$, each with nonempty interior. Collapse the complement of their interiors to a point. As in the pinch construction of [F3], this gives a continuous quotient to $(D_{+}\mathrm{/}\mathrm{\partial }{D}_{+})\vee (D_{-}\mathrm{/}\mathrm{\partial }{D}_{-})\cong S^n\vee S^n$. On the first quotient use an orientation-preserving homeomorphism to the target sphere, taking the collapsed boundary to a chosen basepoint $b$. On the second use an orientation-reversing homeomorphism with the same basepoint image; obtain it by composing an orientation-preserving one with a reflection fixing $b$. Such a reflection has degree $-1$ by [F2].

The two maps agree at the wedge point, so they descend to a continuous map $f:S^n\to S^n$. Each ball quotient already covers the target, so $f$ is surjective and nonconstant. A point $y\mathrm{\ne }b$ has precisely two preimages, one in each ball interior. With the inherited local orientations their local degrees are $+1$ and $-1$: restriction of each oriented quotient homeomorphism to its interior preserves the local generator, and the inserted reflection reverses it. Thus [F1] gives $\mathrm{deg}f=1-1=0$. This refutes constancy, without making any assertion that a degree-zero map cannot be nullhomotopic.