How to prove that w_1(E)=w_1(detE) ? - MathOverflow

How to prove that w_1(E)=w_1(detE) ?

2010-04-17T06:56:16Z

How to prove that the first Stiefel-Whitney class $w_1 (E)$ of a real rank $n$ vector bundle over a manifold M is equal to $w_1(\operatorname{det} E)$, where $\operatorname{det} E$ is the $n$-th wedge power of $E$?

(I want to assume the "axiomatic" definition of Stiefel-Whitney classes, as given e.g. in the book by Milnor and Stasheff).

I have just been asked an analogous question by a younger guy, but I think I could only find a proof starting from a different definition of the $w_i$'s. Perhaps I'm just missing something? Of course, feel free to close it if you find it's to homework-ish for MO standards.

Answer by Thorny for How to prove that w_1(E)=w_1(detE) ?

2010-04-17T09:36:36Z

$E \oplus det E$ is orientable (its structure group $O(n)$ is represented in $SO(n+1)$), so its $w_1$ vanishes; and $w_1(E \oplus det E) = w_1(E) + w_1(det E)$.

Answer by Dan Ramras for How to prove that w_1(E)=w_1(detE) ?

2010-04-17T11:59:39Z

This is really a long comment regarding some of the above discussion.

Hatcher (in his Vector Bundles notes) certainly proves that the characteristic classes defined using Leray-Hirsch satisfy the axioms from Milnor-Stasheff. But w_1 is a more basic object: the axioms specify its value on the tautological bundle over $RP^1$ (= $S^1$ ) and this immediately determines its values on all line bundles (see p. 81 of Hatcher's notes). One can then see that as an element of

$H^1(X; Z/2) = Hom(H_1 X, Z/2) = Hom(\pi_1 X, Z/2)$ ,

$w_1(L)$ simply answers the question: "Along which loops is L trivial?" (Actually, this is true for all bundles, not just lines.) From this point of view, multiplicativity ( $w_1 (L\otimes K) = w_1 L + w_1 K$ ) is a quick exercise (hmmm... what should a homomorphism from a multiplicative group to an additive group be called? Anyway, I just mean it's a homomorphism from the Picard group of line bundles to $H^1$ .). Alternatively, it follows from the H-space structure on $RP^\infty$ defined via the map $RP^\infty\times RP^\infty \to RP^\infty$ classifying $\gamma^1_\infty \otimes \gamma^1_\infty$ . This is spelled out in my notes on vector bundles: (see Lectures 23-25; it's written in the complex case but works the same way in the real case). I couldn't quickly see where Hatcher discusses this point.

Incidentally, my notes also discuss the relationship between orientability and w_1. I've never been crazy about discussions of this point in the literature (e.g. Hatcher states this relationship only for spaces of the homotopy type of a CW complex, although he doesn't seem to use that assumption in his proof).