To make sense of a proof of this nature, it isn't enough to know that there exists a map from $H_d(V, V \backslash \{ y \})$ to $H_d(S^d, S^d \backslash \{ y \})$, and that there exists an isomorphism between $H_d(S^d)$ and $H_d(S^d, S^d \backslash \{ y \})$, and so on. You really also need understand how all of these maps act on the cycles in the various homology groups.
Sometimes (but not always), it is possible to describe how a map acts on cycles in homology groups by identifying the map on homology as the map induced by a continuous function between the relevant topological spaces. Since we usually have a good idea of how to compose continuous functions between topological spaces, this will often enable us to work out how to compose the maps on the homology groups. In certain situations, this will also enable us to prove that certain maps between homology groups are identity maps, or zero maps.
As it happens, all of the maps in your diagram are induced by continuous functions between topological spaces:
The horizontal maps are all induced by $f$ (or restrictions of $f$ to the appropriate spaces).
The top-left vertical map is induced by the identity map $S^d \to S^d$. The same applies to the top-right vertical map. This is just how these maps are defined.
The bottom-left vertical map is induced by the various identity maps $U_i \to U_i$.
The middle-left vertical map is induced by the various inclusion maps $U_i \to S^d$. The bottom-right vertical map is induced by the inclusion map $V \to S^d$. The fact that these maps are induced by the relevant inclusion maps is a part of the statement of the excision theorem, and it is worth noting this!
It's clear that the diagram in your book really is a commutative diagram, because the corresponding maps between topological spaces all commute appropriately!
Each $H_d(U_i, U_i \backslash \{ x_i \} ) $ is isomorphic to $\mathbb Z$, since $$H_d(U_i, U_i \backslash \{ x_i \} ) \cong H_d(S^d, S^d \backslash \{ x_i \}) \cong H_d (S^d) \cong \mathbb Z,$$
where the first equality is by excision and the second equality is by the LES for the pair $(S^d, S^d \backslash \{ x_i \})$.
Therefore, $$H_d (S^d, S^d \backslash \{ x_1, \dots, x_k \} ) \cong \oplus_i H_d(U_i , U_i \backslash \{ x_i \}) \cong \mathbb Z^{\oplus k}.$$
Now let us define the cycle $$(0, \dots, 1, \dots, 0) \in H_d (S^d, S^d \backslash \{ x_1, \dots, x_k \} )$$
(with the $1$ in the $i$th position) to be generator coming from the generator $1 \in H_d(U_i, U_i \backslash \{ x_i \} ) \cong \mathbb Z$.
Here's an important question we must address, if we're to make any progress:
Given a cycle $$(a_1, \dots, a_k) \in H_d (S^d, S^d \backslash \{ x_1, \dots, x_k \} ),$$ is there a simple way to determine the numbers $a_1, \dots, a_k$, if we don't already know them?
And here's my proposed solution:
For each $i \in \{ 1, \dots, k \}$, define a natural map $$p_i : H_d(S^d, S^d \backslash \{ x_1, x_2, \dots, x_k \}) \to H_d(S^d, S^d \backslash \{ x_i \}), $$ to be the map on homology induced by the identity map $S^d \to S^d$.
Then for each $i$, $$ p_i(a_1, \dots, a_k) = a_i \in H_d(S^d, S^d \backslash \{ x_i \}).$$
Let's prove this carefully. First, take $$(1,0,\dots , 0) \in H_d(S^d, S^d \backslash \{ x_1, \dots, x_k \} )$$ and map it to $H_d(S^d, S^d \backslash \{ x_1 \})$ via $p_1$.
Since $(1,0,\dots, 0)$ is itself the image of the generator $1 \in H_d(U_1, U_1 \backslash \{ x_1 \})$, we know that $p_1(1,0,\dots, 0)$ is the image of $1 \in H_d(U_1, U_1 \backslash \{ x_1 \})$ under the composition,
$$ H_d(U_1,U_1 \backslash \{ x_1 \}) \to H_d(S^d, S^d \backslash \{ x_1, \dots, x_k \} ) \to H_d(S^d, S^d \backslash \{ x_1 \}).$$
The first map is induced by the inclusion $U_1 \to S^d$ and the second map is induced by the identity map $S^d \to S^d$, so the composition is induced by the inclusion $U_1 \to S^d$.
But we know that the map $H_d(U_1,U_1 \backslash \{ x_1 \}) \to H_d(S^d, S^d \backslash \{ x_1 \})$ induced by the inclusion $U_1 \to S^d$ is an isomorphism, by excision! So we conclude that $$p_1(1,0, \dots, 0) = 1 \in H_d(S^d, S^d \backslash \{ x_1 \}).$$
Okay, how about we take $$(1,0,\dots , 0) \in H_d(S^d, S^d \backslash \{ x_1, \dots, x_k \} )$$ as before, but this time, we map it via $p_2$ to $H_d(S^d, S^d \backslash \{ x_2 \})$?
The image $p_2(1,0 \dots, 0)$ is the same as the image of $1 \in H_d(U_1, U_1 \backslash \{ x_1 \})$ under the composition,
$$ H_d(U_1,U_1 \backslash \{ x_1 \}) \to H_d(S^d, S^d \backslash \{ x_1, \dots, x_k \} ) \to H_d(S^d, S^d \backslash \{ x_2 \}).$$
But if you think about it, this composition is the same as the composition,
$$ H_d(U_1,U_1 \backslash \{ x_1 \}) \to H_d(U_1,U_1) \to H_d(S^d, S^d \backslash \{ x_2 \}),$$
where the first map is induced by the identity $U_1 \to U_1$ and the second map is induced by the inclusion $U_1 \to S^d$. (Note that since $U_1 \subset S^d \backslash \{ x_2 \}$, the second map really is well-defined.) And why are these two compositions equal? Because both of these compositions are the maps on homology induced by the inclusion map $U_1 \to S^d$!
Of course, $H_d(U_1, U_1) = 0$, so it clear that $$p_2(1,0,\dots 0 ) = 0 \in H_d(S^d, S^d \backslash \{ x_2 \}).$$ This completes the proof of my claim.
Right. Having done all this hard work, we're going to prove that the image of $1 \in H_d(S^d)$ under the top-left map in the diagram is the element $$(1,1, \dots 1) \in H_d(S^d, S^d \backslash \{ x_1, \dots, x_k \} ). $$
I believe this is the part of the proof that you weren't sure about.
By the claim that we just proved, we only have to verify that the image of $1 \in H_d(S^d)$ under the composition
$$ H_d (S^d) \to H_d(S^d, S^d \backslash \{ x_1, \dots, x_k \} ) \overset{p_i} {\to} H_d(S^d, S^d \backslash \{ x_i \})$$
is the element $$1 \in H_d(S^d, S^d \backslash \{ x_i \}).$$
This is straightforward to show. The composition I wrote down is the map on homology induced by the identity map $S^d \to S^d$. But the map $H_d (S^d) \to H_d(S^d, S^d \backslash \{ x_i \})$ induced by the identity map $S^d \to S^d$ is precisely the map appearing in the LES for the pair $(S^d, S^d \backslash \{ x_i \})$, and this map is an isomorphism.
So $1 \in H_d(S^d)$ maps to $1 \in H_d(S^d, S^d \backslash \{ x_i \})$, and we're done.
To finish off, ${\rm deg}f$ is the image of $1 \in H_d(S^d)$ under the map $f_\star$. By the commutative diagram, this is the same as the image of $(1,1,\dots, 1) \in H_d(S^d, S^d \backslash \{ x_1, \dots, x_k \})$ under $f_\star$. And this is the same as the sums of the images of $1 \in H_d( U_i, U_i \backslash \{ x_i \})$ under $f_\star$.
Thus we have shown that
$${\rm deg} f = \sum_i {\rm deg} f_i.$$
Best Answer
For this post, I am thinking of $\mathbb{R}P^n$ modeled by the northern hemisphere of $S^n$ with antipodal equatorial points identified. Also, I am just going to write $f$ instead of $f_n$.
Let $N,S\subseteq S^n$ denote the (closed) northern and southern hemispheres. Let $n\in N$, $s\in S$ be the north and south pole. (For definiteness, $n = (0,0,...,0,1)$ and $s = -n$.
What is $f|_N$? Well, $p|_N$ is the identity (except for at the equator), while $q$ wraps the hemisphere all the way around the sphere. But points on the equatorial $S^{n-1}\subseteq N$ get mapped to $s$.
In other words, we may think of $f|_N$ in hyper polar coordinates as $f(\vec{\theta}, \phi) = 2\phi$. To be clear, $\phi\in [0,\pi]$ measures the angle from $(0,...,1)$ to $x\in S^{n-1}$ and $\vec{\theta} = (\theta_1,....\theta_{n-1})$ with $\theta_1\in[0,2\pi]$, but every other $\phi_i\in[0,\pi],$ is the collection of angle parameters on $S^{n-1}$. (When $n = 2$, $\vec{\theta}$ is the usual $\theta$ in spherical coordinates.)
What is $f|_S$? Well, we first use the antipodal map $a$ to move all these points into the northern hemisphere, then copy $f|_S$. So, $f|_S = f|_N \circ a$. In terms of coordinates, $a(\vec{\theta}, \phi) = (\underline{\vec{\theta}}, \pi-\phi)$ where $\underline{\vec{\theta}} = (-\theta_1, \pi-\theta_2,...,\pi-\theta_{n-1})$.
(Since the anitpodal map has degree $(-1)^{n+1}$, so far this just reproduces your proof that $\deg(f_n) = 1 + (-1)^{n+1}$.)
Thus, we may describe $f$ via $f(\vec{\theta},\phi) = \begin{cases}(\vec{\theta}, 2\phi) & \phi\in[0,\pi/2] \\ (\underline{\vec{\theta}} , 2(\pi-\phi)) & \phi\in[\pi/2,\pi] \end{cases}.$
As a sanity check, this formula clearly gives a continuous $f$ away from $\phi = \pi/2$. But $\lim_{\phi\rightarrow \pi/2^-} (\vec{\theta}, 2\phi) = s = \lim_{\phi\rightarrow \pi/2^+}(\underline{\vec{\theta}}, 2(\pi-\phi))$, so this formula describes a continuous function.
Let's assume $n $ is even. In this language, your question is to find a homotopy between $f$ and a constant map. We will write down this homotopy as a composition of two homotopies. The first homotopy uses the fact that the antipodal map $\vec{\theta}\mapsto \underline{\vec{\theta}}$ is homotopic to the identity because $S^{n-1}$ is an odd dimensional sphere. Suppose $F(\vec{\theta},t)$ is such a homotopy. (For $n=2$, you can use $F(\theta,t) = \theta + t$ with $t\in[0,\pi]$)
We claim that $f_t:=\begin{cases}(\vec{\theta}, 2\phi) & \phi\in[0,\pi/2] \\ F(\underline{\vec{\theta}},t) , 2(\pi-\phi)) & \phi\in[\pi/2,\pi] \end{cases}$ is continuous.
In fact, the argument is just as it was for $f$ above: this is clearly continuous away from $\phi = \pi/2$. And at $\phi = \pi/2$, both formulas limit to $s$, so it is continuous everywhere.
At the end of this homotopy, we get a new map $f_1 = \begin{cases}(\vec{\theta}, 2\phi) & \phi\in[0,\pi/2] \\ (\vec{\theta} , 2(\pi-\phi)) & \phi\in[\pi/2,\pi] \end{cases}.$
Intuitively, this map wraps $N$ all the way around $S^n$ with decreasing lattitude mapping to even more southern lattitudes, then wraps $S$ around the sphere with decreasing lattitude moving northern.
Let's use a final homotopy to get the constant map.
The homotopy here is $G(\vec{\theta}, \phi, t) = \begin{cases}(\vec{\theta}, 2t\phi) & \phi\in[0,\pi/2] \\ (\vec{\theta} , 2t(\pi-\phi)) & \phi\in[\pi/2,\pi] \end{cases}.$ Once again, we need to check that this is continous, and again, this is obvious away from $\phi = \pi/2$.
But when $\phi = \pi/2$, both the top and bottom map send $(\vec{\theta},\phi)$ to $(\vec{\theta}, \pi t)$, so $G$ is continuous. Finally, simply note that $G(\vec{\theta}, \phi, 0) = (\vec{\theta},0) = n$, so is constant.