$\mathbb{R}P^3$ is orientable, but $\mathbb{R}P^2$ is not

Question

Def: Let $U, V \subset \mathbb{R}^n$ be open and $f: U \to V$ a homeomorphism. We say $f$ is orientation preserving if for all $x \in U$ the composite $$ H_n(\mathbb{R}^n, \mathbb{R}^n \setminus 0) \xrightarrow{tr} H_n(\mathbb{R}^n, \mathbb{R}^n \setminus x) \xrightarrow{exc} H_n(U, U \setminus x) \xrightarrow{f_*}H_n(V, V\setminus f(x)) \xrightarrow{exc} H_n(\mathbb{R}^n, \mathbb{R}^n \setminus f(x)) \xrightarrow{tr} H_n(\mathbb{R}^n, \mathbb{R}^n \setminus 0)$$ is the identity map. (Here tr denotes the map induced by translation of $\mathbb{R}^n$ and exc denotes the excision isomorphism.)

Lemma: A manifold $M$ is orientable iff it admits an atlas with transition maps orientation preserving homeomorphisms of open subsets of $\mathbb{R}^n$.

I am trying to show that $\mathbb{R}P^3$ is orientable using this lemma. So I constructed an atlas with four charts $U_1 = \{ [1:y:z:w] \in \mathbb{R}P^3 \} \xrightarrow{\phi_1} \mathbb{R}^3$ via $[1:y:z:w] \mapsto (y,z,w)$,

$U_2 = \{ [x:1:z:w] \in \mathbb{R}P^3 \} \xrightarrow{\phi_2} \mathbb{R}^3$ via $[x:1:z:w] \mapsto (x,z,w)$

and similarly for $U_3$ and $U_4$.

Now we would like the transition maps $\phi_i \phi_j^{-1}$ to be orientation preserving. This is where the problems start for me.

Let's check that $f=\phi_2 \phi_1^{-1}$ is orientation preserving. Let $(y,z,w) \in \mathbb{R}^3$ be arbitrary. Then $f(y, z, w)=(1/y, z/y, w/y)$.

However, the same proof should break down for $\mathbb{R}P^2$, but I can't see where this happens, so I know something is wrong with this "proof".

Answer 1

You have to show that the maps $\phi_i \phi_j^{-1}$ are orientation preserving. For $i=2$,$j=1$ and your notation, this is: $$(y,z,w) \mapsto (\frac{1}{y},\frac{z}{y},\frac{w}{y})$$ where we have to restrict to the set where $y\neq 0$. This changes the orientation of $\mathbb{R}^3\setminus \{ y=0\} $ (in particular, it is not the identity!).

One thing you could try to construct an atlas would be the following: Pick an atlas of orientation preserving charts of $S^3$ such that all domains of the charts are contained in a hemisphere. Since the antipodal map on $S^3$ is orientation-preserving, this will give you an atlas of charts such that on the intersection of two charts the transition diffeomorphisms are orientation-preserving.