Vector bundle of dimension $\leqslant n$ on $n$-connected space is trivial I wonder whether any vector bundle of dimension $\leqslant n$ on an $n$-connected CW-complex is trivial? It seems that,


*

*the complex can be given cellular structure with exactly one 0-dimensional cell and no other cells of dimension $\leqslant n$,

*there is a inner product on the vector bundle by the paracompactness of CW-complexes,

*the corresponding $S^{n-1}$-fiber bundle, by some obstruction theory argument (I am not sure which one, but it should use the fact that all maps $S^{\leqslant (n-1)} \to S^n$ are homotopic), has a section,

*and its orthogonal complement has smaller dimension, so allowing the proof by induction.


For $n=1$ it is true: any covering (with two sheets) over simply-connected space is trivial. But I have not found the general statement anywhere, so it there an error?

Your proposed argument fails at the third step. It is just not true that if $X$ is $n$-connected then every $S^{n-1}$-bundle over it is trivial. 
The desired statement is not even true rationally: the rational homotopy groups of $BO(n)$ are not hard to calculate and in particular for $n \ge 3$,
$$\pi_{4n-4}(BO(2n)) \otimes \mathbb{Q} \cong \mathbb{Q}$$ 
so it follows that there are countably many nontrivial $2n$-dimensional real vector bundles over $S^{4n-4}$, which moreover can be distinguished by their Pontryagin classes $p_{n-1}$, and this gives an infinite family of counterexamples to the desired statement.
