Cellular Boundary Formula In Hatcher's book we find, when computing the boundary maps of cellular homology,
Cellular Boundary Formula: $d_n(e^{n}_\alpha)=\sum_\beta d_{\alpha\beta}e^{n-1}_{\beta} $ where $d_{\alpha\beta}$ is the degree of the map $S^{n-1}_\alpha \rightarrow X^{n-1}\rightarrow S^{n-1}_\beta$ that is the composition of the attaching map of $e^{n}_\alpha$ with the quotient map collapsing $X^{n-1}- e^{n-1}_\beta $ to a point.
My question is:
When we identify $X^{n-1}/(X^{n-1}- e^{n-1}_\beta)$ with $S^{n-1}_\beta$, we are choosing a homeomorphism. Doesn't this choice affect the degree of the map in question? 
Essentially (as I see it), we must choose a quotient map from $D^{n-1}$ to $S^{n-1}$ which identifies $S^{n-2}$ to a point. 

 

The left vertical arrow in the upper diagram is, up to the isomorphism described in the lower diagram, the inclusion of a direct $\Bbb Z$-summand into the direct sum. Composing this inclusion with $d_n$ and subsequently with the map to $\tilde H_{n-1}(S^{n-1}_\beta)$, which algebraically is the projection onto the direct summand for $\beta$, we get the composition $\Delta_{\alpha\beta*}\partial$. If we change the homeomorphism $h_β:D_β/∂D_β\to S_β$, so that the "typical" generator of $H_{n-1}(D_β,∂D_β)$ gets now mapped to the negated "typical" generator of $H_{n-1}(S_β)$, then each $e_β$ among the images of the cells under $d_n$ is replaced by $-e_β$. However, each $e_β$ among the generators of the kernel of $d_{n-1}$ would also have to be replaced by $-e_β$ since the value $d_{n-1}(e_β)$ has been negated by changing the homeomorphism. So overall nothing would change.
