How to prove that w_1(E)=w_1(detE) ? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T16:35:14Z http://mathoverflow.net/feeds/question/21649 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/21649/how-to-prove-that-w-1ew-1dete How to prove that w_1(E)=w_1(detE) ? Qfwfq 2010-04-17T06:56:16Z 2010-04-17T13:04:11Z <p>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 $E$?</p> <p>(I want to assume the "axiomatic" definition of Stiefel-Whitney classes, as given e.g. in the book by Milnor and Stasheff).</p> <p>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.</p> http://mathoverflow.net/questions/21649/how-to-prove-that-w-1ew-1dete/21654#21654 Answer by Thorny for How to prove that w_1(E)=w_1(detE) ? Thorny 2010-04-17T09:36:36Z 2010-04-17T09:36:36Z <p>$E \oplus det E$ is orientable (its structure group $O(n)$ is represented in $SO(n+1)$), so its $w_1$ vanishes; and $w_1(E \oplus det E) = w_1(E) + w_1(det E)$.</p> http://mathoverflow.net/questions/21649/how-to-prove-that-w-1ew-1dete/21660#21660 Answer by Dan Ramras for How to prove that w_1(E)=w_1(detE) ? Dan Ramras 2010-04-17T11:59:39Z 2010-04-17T13:04:11Z <p>This is really a long comment regarding some of the above discussion.</p> <p>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 <code>$RP^1$</code> (= <code>$S^1$</code>) 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 </p> <p><code>$H^1(X; Z/2) = Hom(H_1 X, Z/2) = Hom(\pi_1 X, Z/2)$</code>, </p> <p><code>$w_1(L)$</code> 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 (<code>$w_1 (L\otimes K) = w_1 L + w_1 K$</code>) 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 <code>$H^1$</code>.). Alternatively, it follows from the H-space structure on <code>$RP^\infty$</code> defined via the map <code>$RP^\infty\times RP^\infty \to RP^\infty$</code> classifying <code>$\gamma^1_\infty \otimes \gamma^1_\infty$</code>. 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. </p> <p>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).</p>