Geometric picture of Stiefel-Whitney class of a tangent bundle?

Question

The first Stiefel-Whitney class $w_1$ of a tangent bundle of a manifold $M$ has a simple geometric picture: if there is a loop that the orientation of the tangent space reverses, then $w_1\neq 0$ and the evaluation of $w_1$ on the loop is $1$. Do we have a simple geometric picture for the second Stiefel-Whitney class $w_2$ (and more generally for $w_n$)?

Answer 1

An answer to your question on this can be found at Hatcher's book Vector Bundles and K-theory(http://www.math.cornell.edu/~hatcher/VBKT/VB.pdf) page 75 in terms of obstruction classes. I think Hatcher's discussion is enlightening and I encourage you to read it yourself. I do not think I can present it better than he did.

This book is a standard reference book for characteristic classes, so it might be helpful for you to read other parts as well. It has has a strong geometric flavor, which you might like.