Direct proof of decomposition of real vector bundle of odd degree into the direct sum of a trivial bundle and another of even degree

Question

The real splitting principle tells us that when taking a real, oriented vector bundle of odd dimension $\zeta$ over a manifold $M$ you can always write $\zeta$ as $\tilde{\zeta} \oplus \varepsilon^1$, where $\dim\tilde{\zeta} = \dim \zeta - 1$ and $\varepsilon^1$ is the trivial bundle over $M$, that is, $\varepsilon^1 = M \times \mathbb{R}$ with the projection on the first factor. Is there an easy way to show this directly?

Thanks a lot!

Answer 1

This is not what I would call the real splitting principle, and also it is not true. What is true is the following:

• A real oriented vector bundle $V$ splits as above iff it has a nowhere vanishing section iff its Euler class $e(V)$ vanishes. (Edit, 4/30/21: As Georges says in the comments, this is incorrect; whoops! Fortunately what is true, and what is needed for the rest of the argument in this answer, is that if an oriented vector bundle has a nowhere vanishing section then its Euler class vanishes.)
• If $V$ is odd-dimensional, then $V$ is isomorphic as an oriented vector bundle to the same vector bundle but with the opposite orientation, and since reversing the orientation reverses the Euler class, we have $e(V) = - e(V)$, or $2e(V) = 0$.

But it is possible for the Euler class of a real oriented odd-dimensional vector bundle to be a nontrivial $2$-torsion class. Since $e(V) \equiv w_n(V) \bmod 2$, it suffices to exhibit a $3$-dimensional real orientable vector bundle whose top Stiefel-Whitney class $w_3$ does not vanish. By the splitting principle, if an example exists, then an example exists which is a direct sum of three line bundles. The Stiefel-Whitney classes $\alpha, \beta, \gamma \in H^1(-, \mathbb{Z}_2)$ of these line bundles must satisfy

$$\alpha + \beta + \gamma = 0$$

(for orientability) and

$$\alpha \beta \gamma \neq 0$$

(because this is the top Stiefel-Whitney class $w_3$ and we don't want it to vanish). We can find classes in $H^1(-, \mathbb{Z}_2)$ with this property on $M = \mathbb{RP}^2 \times \mathbb{RP}^2$. By the Kunneth formula,

$$H^{\bullet}(M, \mathbb{Z}_2) \cong \mathbb{Z}_2[\alpha, \beta]/(\alpha^3 = \beta^3 = 0)$$

where $\alpha$ and $\beta$ are the Stiefel-Whitney classes of the tautological bundles $L_{\alpha}, L_{\beta}$ on the first and second copy of $\mathbb{RP}^2$ respectively. Take $\gamma = \alpha + \beta$. The resulting direct sum of line bundles

$$V = L_{\alpha} \oplus L_{\beta} \oplus (L_{\alpha} \otimes L_{\beta})$$

is a $3$-dimensional real orientable (because $\alpha + \beta + \gamma = 0$ vanishes by construction) vector bundle with top Stiefel-Whitney class

$$w_3(V) = \alpha \beta (\alpha + \beta) = \alpha^2 \beta + \alpha \beta^2 \neq 0$$

as desired.