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.