I am reading up on Stiefel-Whitney classes $w_k(T_M)$ on $m$-manifolds; $k=1,2,\dots,m$, which are described as obstacles to extending a trivialization of a bundle, from the $k$-skeleton (assume manifolds are nice-enough to admit a cell-decomposition); for $k=1$, it is just the obstruction to orientability (we just define $w_1$ over embedded circles $C$, and define $w_1(C)=0$ if $C$ preserves orientation, and $w_1(C)=1$ otherwise. A nowhere-zero vector field is equivalent to orientability)
Question: anyone know of an example of a manifold $M$, say for $\dim \leq4$ that admits a trivialization over its $1$-skeleton (i.e., $M$ is orientable), but so that the trivialization does not extend to the $2$-skeleton? (and, is this trivialization over the $1$-skeleton a line bundle as the restriction to the $1$-skeleton of the $\mathbb R^4$-bundle?)
Even better if there are examples of trivialization over $1$- and $2$-skeleton, but not $3$-skeleton, etc.
Thanks in Advance,