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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 ${\Bbb R}P^1$ (= $S^1$) 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

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

$w_1(L)$ 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 ${\Bbb R}P^\infty$ defined via the map ${\Bbb R}P^\infty\times {\Bbb R}P^\infty \to {\Bbb R}P^\infty$ 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).

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 ${\Bbb R}P^1$ (= $S^1$) 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

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

$w_1(L)$ 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 ${\Bbb R}P^\infty$ defined via the map ${\Bbb R}P^\infty\times {\Bbb R}P^\infty \to {\Bbb R}P^\infty$ classifying $\gamma^1_\infty \otimes \gamma^1_\infty$. This is spelled out in my notes on vector bundles (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).

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 ${\Bbb R}P^1$ (= $S^1$) 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

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

$w_1(L)$ 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 ${\Bbb R}P^\infty$ defined via the map ${\Bbb R}P^\infty\times {\Bbb R}P^\infty \to {\Bbb R}P^\infty$ 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).

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 ${\Bbb R}P^1$ (= $S^1$) 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

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

$w_1(L)$ 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 ${\Bbb R}P^\infty$ defined via the map ${\Bbb R}P^\infty\times {\Bbb R}P^\infty \to {\Bbb R}P^\infty$ 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).

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).

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 ${\Bbb R}P^1$ (= $S^1$) 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

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

$w_1(L)$ 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 ${\Bbb R}P^\infty$ defined via the map ${\Bbb R}P^\infty\times {\Bbb R}P^\infty \to {\Bbb R}P^\infty$ 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).