Here are some statements that I wish to understand more deeply, whose truth value I want to check, and to determine under which criteria they are valid.
Consider the Stiefel manifold $V_k(R^n)$ of orthogonal $k$-frames in $R ^n$. To the frame bundle, one associates a bundle where $n$ is the dimension of $X$.
The Stiefel manifold can be understood as an iterated sphere fibration. The choice of the first unit vector is a point on $S^{n−1}$. Then we choose a unit vector on the hyperplane normal to that vector, whih can be thought of as the tangent space to $S^{n−1}$ at that point. Thus, $V^k(R^d)$ is the k-tuply iterated unit tangent bundle of $S^{d−1}$.
By choosing all but the last vector, our remaining choice is a point on $S^{d−k}$. This is the smallest sphere in the fibration, we conclude $π^{d−k}V_k(R^d) = π_{d−k}(S^{d−k}) = \mathbb{Z}$ and all the lower ones are zero.
By Hurewicz theorem, $H^{d−k}(V_k(R^d))$ is generated by this element. If we send the generator to $−1$, we get an element of $H^{d−k}(V_k(R^d), \mathbb{Z}_2)$.
The transgression of this element from the associated Stiefel bundle down to $X$ is $w_{n−k+1}$.
If the above all are true, have we used the above facts to explain this statement obstruction?
The class $w_{n−k+1}$ is the obstruction to finding a section of this bundle. If $w_n = 0$ then $X$ can have non-vanishing vector fields.