Degree is not defined by top homology for self maps of s zero

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)

For $n\ge 1$, ${\tilde{H}}_k(S^n;G)$ is $G$ for $k=n$ and $0$ otherwise. For $S^0$, ${\tilde{H}}_0(S^0;G)\cong G$ and all other reduced groups vanish. Thus $H_0(S^n;G)\cong G$ for $n\ge 1$, whereas $H_0(S^0;G)\cong G\oplus G$. (Homology of spheres)

For an unreduced theory $h$ and a nonempty based CW space $(X,x_0)$ with $x_0$ a vertex, set ${\tilde{h}}_n(X)=\ker (h_n(X)\stackrel{p_{\ast }}{\to }{h}_n(\ast ))\mathrm{.}$ The basepoint inclusion $s$ satisfies $p\circ s=\mathrm{i}\mathrm{d}$ and splits this augmentation. The underlying ordinary theory is as in def-unreduced-homology-theory-on-cw-pairs. Independently, a reduced ordinary theory on based CW spaces consists of homotopy-invariant covariant functors ${\tilde{h}}_n$, natural suspension isomorphisms $\sigma :{\tilde{h}}_n(X)\to {\tilde{h}}_{n+1}(\mathrm{\Sigma }X)$, exact cofiber sequences, and arbitrary wedge additivity. More explicitly, for every based CW inclusion $A\hookrightarrow X$, ${\tilde{h}}_n(A)\to {\tilde{h}}_n(X)\to {\tilde{h}}_n(X\mathrm{/}A)$ is exact; the boundary in the extended sequence is the cofiber map to $\mathrm{\Sigma }A$ followed by ${\sigma }^{-1}$. The suspension here is reduced suspension. The dimension axiom is ${\tilde{h}}_n(S^0)=0$ for $n\mathrm{\ne }0$, with ${\tilde{h}}_0(S^0)=G$. Wedge additivity includes the empty wedge $\ast$ and gives ${\tilde{h}}_n(\ast )=0$. The empty space is not a based object. If its reduced groups are mentioned, this library uses ${\tilde{H}}_n(\mathrm{\varnothing };G)=0$ in all degrees, as in def-zero-simplex-augmentation-and-reduced-singular-homology. The augmented-chain convention ${\tilde{H}}_{-1}(\mathrm{\varnothing };G)=G$ is a different extension and is not used here. (Reduced homology theory and augmentation)