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This question is motivated by the vector field question I asked recently. Panagiotis Konstantis answered this question for odd manifolds and I am trying to figure out the even case.

Let $M$ be a smooth oriented manifold (without boundary) of even dimension $2k$ with $k \geq 2$. Steenrod showed that the primary obstruction for lifting the tangent bundle $\tau\colon M \rightarrow BO(2k)$ along the fibration $$V_2(\mathbb{R}^{2k}) \rightarrow BO(2k-2) \rightarrow BO(2k)$$ is $$ \beta^*(w_{2k-2}) \in H^{2k-1}(M; \pi_{2k-2}(V_2(\mathbb{R}^{2k})) = H^{2k-1}(M; \mathbb{Z})\,,$$ where $\beta^*$ is the Bockstein operator and $w_{2k-2}$ is the $(2k-2)$th Stiefel-Whitney class of $M$.

Now Theorem 2 of Massey's "On the Stiefel-Whitney classes of a manifold II" paper says that this class vanishes when $M$ is closed. Can we say the same for open $M$? If the cohomology class had field coefficients, we could argue the vanishing as in this answer and the comments under it. But here the class is integral. Can we salvage the situation using other properties the problem has (such as $\beta^*(w_{2k-2})$ being 2-torsion)?

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Your question is the following:

Does $W_{2k-1}$ vanish on an (open) orientable $2k$-manifold?

I don't know the answer in general, only for $k = 1, 2, 3$.

  • We always have $W_1 = \beta^*(w_0) = \beta^*(1) = 0$, so the answer is yes for $k = 1$.

  • Note that $W_3 = \beta^*(w_2)$ is the obstruction to the existence of a spin$^c$ structure on an orientable manifold. As every orientable four-manifold is spin$^c$ (see this note by Teichner and Vogt), the answer is also yes for $k = 2$.

  • In this paper, Aleksandar Milivojevic and I showed that $W_5$ is the primary obstruction to the existence of a spin$^h$ structure on an orientable manifold (Corollary 2.6). As every orientable six-manifold is spin$^h$ (Corollary 3.10), the answer is also yes for $k = 3$.

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