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I am having some trouble with Bott–Tu's definition of orientability of a sphere bundle. I've pasted the definition in as a picture below. In particular, in the first sentence of the second paragraph, they say that "a generator of the top cohomology of a fiber is an $n$-form with total integral 1." But, since $H^n(S^n)=\mathbb R$, shouldn't it be enough that the total integral is just nonzero? In particular, couldn't we have a problem where $[\sigma_\alpha]=2[\sigma_\beta]$, for example?

I also don't quite understand why an $S^0$-cover is orientable if and only if the total space contains two connected components (note that we assume the base space is connected). In the backwards direction, I understand it: Basically, each $E|_{U_\alpha}$ is disconnected, so must contain a piece in each connected component. Thus each fiber contains one point in each component. So just fix one component; for each fiber, consider the generator (a 0-form) which is 1 in that component and 0 on the other component. Could someone explain or give me a hint for the forward direction though?

I might be missing something obvious with regards to both questions; I've put down this book for a few weeks, and I think I've forgotten some of the stuff from earlier!

enter image description here

boink
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    I don't have time to right a complete answer, but for Q1 you're right that having total integral non-zero is sufficient. This is related to the fact that $\mathbb{R}$ is a field, so any element generates it (as an $\mathbb{R}$-module). On the other hand, there are other contexts (e.g., singular cohomology), where, e.g., $H^n(S^n)\cong \mathbb{Z}$ and now not all non-zero elements are created equally. Regarding Q2:, and element of $H^0(S^0)$ is really just a choice of one of the two fiber points. If you can make this choice consistently, your $S^0$-bundles has a section, so it is trivial. – Jason DeVito - on hiatus Jan 05 '23 at 13:55
  • @JasonDeVito Thanks, I see. So for the first question, I guess I can just take a "consistent" set of generators to refer to ones that are of the same sign? – boink Jan 05 '23 at 21:16

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