I am interested to know the general condition for a real vector bundle $V$ over an unorientable manifold $X$ ($X$ can be either 4d or 5d) to be stably trivial.
The case I've heard of is when $X$ is 4d unorientable, then $V$ is stably trivial iff $w_1(V)=w_2(V)=w_4(V)=0$ all vanish.
Then what is the corresponding "iff" condition when $X$ is 5d unorientable?
(I am also interested to know a reference where the proof for 4d $X$ is presented. )
Any comments, references are welcome.
A stable bundle $V$ on $X$ is trivial if the map $X \to BO$ it classifies is nullhomotopic. In order to analyse maps from a finite-dimensional complex into $BO$, we can study the Whitehead tower of $BO$ and its obstruction theory.
Here is the Whitehead tower of $BO$, up to dimension 8:
\begin{array}{ccc} \tau_{\geq 8} BO & \xrightarrow{\frac{p_2}{6}} & K(\mathbb{Z}, 8) \\ \downarrow \\ \tau_{\geq 4} BO & \xrightarrow{\frac{p_1}{2}} & K(\mathbb{Z}, 4) \\ \downarrow \\ \tau_{\geq 2} BO & \xrightarrow{w_2} & K(\mathbb{Z}/2, 2) \\ \downarrow \\ BO & \xrightarrow{w_1} & K(\mathbb{Z}/2, 1) \end{array}
If $X$ is a complex of dimension 4 or less, then a map $X \to BO$ is null iff the maps \begin{align*} &w_1: X \to K(\mathbb{Z}/2, 1) \\ &w_2: X \to K(\mathbb{Z}/2, 2) \\ &\frac{p_1}{2}: X \to K(\mathbb{Z}, 4) \end{align*} are null. This gives a characterization of stably trivial bundles in terms of characteristic classes.
Furthermore, this also shows that there are no extra conditions needed when $X$ is 5-dimensional, or 6-dimensional, or 7-dimensional. When $X$ is 8-dimensional, however, there is an additional obstruction given by $\frac{p_2}{6}$ in order for the bundle to be stably trivial.