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Let $E \to B$ be real vector bundle. Consider the homomorphism $\pi_1(B) \to \mathbb{Z}_2$ which assign $0$ or $1$ to each loop according to whether orientations of fibers are preserved or not. Since $\mathbb{Z}_2$ is abelian this homomorphism factors through $H_1(B)$ therefore it gives a class in $H^1(B,\mathbb{Z}_2)$.

How to prove that this class is exactly the first Stiefel_Whitney class?

EDIT: My preferred definition of S-W classes is via axiomatic approach: naturality, Whitney formula and normalization

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  • $\begingroup$ math.stackexchange.com/questions/1820394/… $\endgroup$ – Lorenzo Jul 5 '17 at 15:59
  • $\begingroup$ What is your initial definition of Stiefel Whitney first class ? $\endgroup$ – R. Alexandre Jul 5 '17 at 16:02
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    $\begingroup$ Consider how the given class behaves under direct sum and pullbacks, then reduce the problem (e.g., via the splitting principle) to the case of the tautological bundle over $\mathbb{RP}^\infty$. $\endgroup$ – anomaly Jul 5 '17 at 16:40
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    $\begingroup$ It has the same functorality so it suffices to check the universal examples, namely the universal vector bundles on $BO(n)$. Here you can just check that there's only one nonzero class in that cohomology group so it must be $w_1$ if it's nonzero. $\endgroup$ – Qiaochu Yuan Jul 5 '17 at 16:40
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What is your preferred definition of the first Stiefel Whitney class?

I'm more comfortable with classifying spaces (this is the only definition I can produce for Stiefel Whitney classes right now), and so I'd prefer to think of your description as whether or not the classifying map $f : B \to BO(n)$ factors through $BSO(n)$, i.e. whether or not the bundle is orientable (i.e. whether or not the transition maps can be taken to be determinant one).

Because of the short exact sequence $0 \to SO(n) \to O(n) \to \mathbb{Z}/2Z \to 0$, where the last map is the determinant, we get a fibration $B(SO(n)) \to B(O(n)) \to B(Z/2Z) = \mathbb{RP}^{\infty}$. The bundle is orientable iff the classifying map can be lifted to $SO(n)$ (up to homotopy), iff it can be homotoped to live in a fiber, iff the induced map to $\mathbb{RP}^{\infty}$ is null homotopic (homotopy lifting property for Serre fibrations), iff the determinant line bundle is orientable, iff the principle $O(1)$ bundle (2 sheeted cover) associated to the determinant line bundle is trivial. This is exactly keeping track of the $\pi_1(B)$ action on the orientation of a fiber.

The last bunch of equivalences can be used to solve the problem. The point is the the map from X to $RP^{\infty}$ is nullhomotopic iff the induced map on $H_1$ is trivial iff the induced map on $\pi_1$ is trivial, iff it is lifts to the cover which is $S^{\infty}$, and in particular contractible. (We use that $\pi_1(RP^{\infty})) = \mathbb{Z}/2$.) The induced map on $H_1$ is dual to the induced map on $H^1$ (everything is over a field*), so you can conclude from this, since the first Stiefel Whitney class is the pullback of the generator $w_1$ of $H^1(B(O(n))$ under the classifying map.

(*This gave me pause, so let me explain. This is pure homological algebra. We have two chain complexes of k-vector spaces, $C_*$ and $D_*$, an a map $f : C \to D$. We have a functor $F : k-Vect \to k-Vect$, $Hom(\_, k)$, which is exact. Then the induced map of $H_* (F(C_*))) \to H_*(F(D_*))$ is $F$ of the induced map $H_*(C_*) \to H_*(D_*)$. The point is that exact functors "commute" with taking homology of a complex.)

I'm assuming that $X$ is a CW-complex, because at some point I use a result about Serre fibrations, but probably this can be done away with. Let me know if you have questions, my writing here is hardly masterly exposition - I'm trying to understand this material as well.

This is very close to the content of The first Stiefel-Whitney class is zero if and only if the bundle is orientable but I think I address your question (about the action of $\pi_1$) more directly, and add some more detail.

If this is all nonsense to you (or other reader), I learned these ideas (except the mistakes) from these notes, which I cannot advertise enough: https://www3.nd.edu/~mbehren1/18.906/prin.pdf

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