I am reading chapter 11 of Differential Forms in Algebraic Topology by Bott & Tu. The argument in the following paragraph seems to imply that the Euler class of any oriented sphere bundle would vanish. I believe that it is wrong, since Example 11.18 gives a counterexample, but I don't know why. Can somebody help me find the error?
In the middle of page 119 of the book, the authors said that
In sum, there is a global form that restricts to a generator on each fiber if and only if
(a) $E$ is orientable, and
(b) the Euler class $e(E)$ vanishes.
It can be concluded that if $E$ is any oriented sphere bundle and there is a global form that restricts to a generator on each fiber, then the Euler class $e(E)$ vanishes. But the global angular form constructed in page 121-122 really is a global form that restricts to a generator on each fiber. It can be constructed on any oriented sphere bundle. This implies that the Euler class of any oriented sphere bundle would vanish.