A map of nonzero degree between spheres is surjective

Given: The objects and hypotheses in the statement above.

Let $n\ge 1$. Choose a generator $[S^n]$ of $H_n(S^n;\mathbb{Z})\cong \mathbb{Z}$, using cor-homology-of-spheres. For a continuous self-map $f:S^n\to S^n$, its degree is the unique integer satisfying $f_{\ast }[S^n]=\mathrm{deg}(f)[S^n]\mathrm{.}$ The induced map is furnished by prop-relative-homology-is-functorial-for-maps-of-pairs with empty subspaces. Replacing the same generator in source and target by its negative does not change the integer. For a map between separately oriented copies of $S^n$, use their separately specified generators; reversing just one orientation changes the sign. The unreduced definition here is restricted to $n\ge 1$. (Degree of a self map of an oriented sphere)

If $X$ is a nonempty contractible topological space, then for every $n\ge 0$ and every abelian group $G$, $H_n^{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}(X;G)\cong H_n^{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}(\ast ;G),$ where $\ast$ denotes a one-point space. (Contractible nonempty spaces have the homology of a point)

For any abelian group $G$, $H_0(\ast ;G)\cong G$ and $H_n(\ast ;G)=0$ for every integer $n\mathrm{\ne }0$. For every set-indexed family of pairs the canonical map ${⨁}_{\alpha }{H}_n(X_{\alpha },A_{\alpha };G)⟶H_n({⨆}_{\alpha }{X}_{\alpha },{⨆}_{\alpha }{A}_{\alpha };G)$ is an isomorphism. Together with the structural axioms, singular homology is an ordinary theory with coefficient group $G$. (Singular homology satisfies dimension and arbitrary additivity)