I understand this question has been asked once already, but my issue is with understanding example 2.42 from Hatcher's Algebraic Topology. He starts with,
"$\mathbb{R}P^n$ has a CW structure with one cell $e_k$ in each dimension $k\leq n$, and the attaching map for $e_k$ is the 2 sheeted covering projection $φ : S^{k−1}→\mathbb{R}P^{k−1}$. To compute the boundary map $d_k$ we compute the degree of the composition $S^{k−1}\xrightarrow{φ}\mathbb{R}P^{k−1}\xrightarrow{q}\mathbb{R}P^{k−1}/\mathbb{R}P^{k−2}=S^{k−1}$, with q the quotient map."
He then continues to state, and this is the part I have an issue with,
"The map $qφ$ is a homeomorphism when restricted to each component of $S^{k−1} - S^{k−2}$, and these two homeomorphisms are obtained from each other by precomposing with the antipodal map of $S^{k−1}$, which has degree $(−1)^k$. Hence $\deg(qφ)=\deg(\mathbb{1})+deg(−\mathbb{1})=1+(−1)^k$, and so $d_k$ is either 0 or multiplication by 2 according to whether $k$ is odd or even."
I understand that if we restrict to each component of $S^{k−1} - S^{k−2}$, call these $B_+$ and $B_-$, we will get the identity map and the antipodal map (as seen in 6.5 here). And if I am correct since $S^{k−2}$ is just the pre-image of $\mathbb{R}P^{k-2}$ under $φ$, it is mapped to a single point. But how do we deduce that the degree of $qφ$ is just the sum of the degree of the identity and the antipodal map?
One definition of "degree" (for maps between compact oriented $n$ manifolds) is this:
Pick any codomain point $P$. Adjust (by small deformation, using a general position kind of argument) the map $f$ so that (1) $f^{-1}(P)$ is a finite set, and (2) on $f^{-1}(P)$, $df$ has maximal rank (i.e., for each point $Q_i$ in the preimage, $df(Q_i)$ is an isomorphism of vector spaces). Then pick an oriented basis $\mathbf b$for each point $Q_i$ in $f^{-1}(P)$, and look at $df(Q_i)(\mathbf b)$ and check whether it's positively ($+1$) or negatively ($-1$) oriented in the codomain. Sum up the results to get the degree.
Hatcher is using this for a typical non-$RP^{k-2}$ point in the codomain, and finds two preimage points, one where the map has degree $+1$, and the other where it has degre $\pm 1$, and so on.