We can define the degree, $d$ of a continuous map $f:S^n \to S^n$ through the induced map, $f_*$, in homology: $x \mapsto dx$.
Now consider a map $S^{n-1} \times S^{n-1} \to S^{n-1}$, and let $y \in S^{n-1}$. Let $\alpha$ denote the degree of $g$ restricted to $S^{n-1} \times y$ and $\beta$ the degree of $g$ restricted to $y \times S^{n-1}$. Show that $\alpha$ and $\beta$ are independent of the choice of $y$.
I am not really sure what I need to prove here! If the degree $\alpha$ is simply given by the induced map $H_{n-1}(S^{n-1} \times y) \to H_{n-1}(S^{n-1})$, then there is really nothing to prove since multiplying by a point $y$ does not change the homology.
I am sure that is not exactly what is required. So, any hints for what to do in this question?