Orientability of $P_{\bf R}T{\bf RP}^{2n}$

Question

I know the following fact : (1) $ {\bf RP}^{2n}$ is non-orientable. (2) $ {\bf RP}^{2n-1}$ is orientable. (3) $P_{\bf R}T{\bf RP}^{2n}$ is orientable. (4) $P_{\bf R}T{\bf RP}^{2n+1}$ is non-orientable.

First two things are proved by the following statement :

Antipodal map : $S^n\rightarrow S^n$ has degree $(-1)^{n+1}$

I wonder how to prove remaining two things. These are ${\bf Z}_2\times {\bf Z}_2$-quotients of unit sphere bundle over a sphere, i.e., $T_1S^k$. So in my thought we can view as ${\bf RP}^{k-1}\times {\bf RP}^k$ when we are considering orientation. Whence two are non-orientable. What is wrong ?

Thank you in advance.

[Definition]--------------------------------------------------------------

$P_{\bf R} T{\bf RP}^n$ is projective tangent bundle, that is $T_1{\bf RP}^n/{\bf Z}_2$, where $T_1M$ is a unit sphere bundle over $M$ which is in $TM$

[Reference]--------------------------------------------------------------

I want to read "Manifolds with positive sectional curvature almost everywhere - B. Wilking" The above examples are in it. In here $P_{\bf R}T{\bf RP}^{2n+1}$ is stated as non-orientable. But $P_{\bf R}T{\bf RP}^{2n}$ is not listed in counter examples so that I concluded that it is orientable.

Answer 1

I suspect you mean $\mathbb{RP}^{2n/2n+1}$ instead of $\mathbb{R}^{2n/2n+1}$ in the first two cases, where the orientability is tackled by the top degree homology class or by Stiefel-Whitney class. Otherwise this would not make much sense unless you mean the exotic $\mathbb{R}^{4}$.

For the later suspect you mean the projectivization on each fibre of $T(\mathbb{RP}^{2n/2n+1})$ as a manifold, where each fibre now become $P(\mathbb{R}^{2n/2n+1})$. But this bundle is in general not trivializable. In dimension higher you may encounter parallizable $\mathbb{RP}^{n}$s(there should be only 3 cases), and then this might make some sense.

For your question since the $P(T(\mathbb{RP}^{2n}))$ is the double cover of $S(T(\mathbb{RP}^{2n}))$, can you work out the invariants for $S(T(\mathbb{RP}^{2n}))$? The sphere bundle should be easier to tackle as you can form appropriate fibrations, etc with it.