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It is a well known fact that (isomorphism classes of) princial $\mathbb{S }^1$-bundles over a base space $B$ are classified by $B$'s second integral cohomology, $H^2(B;\mathbb{Z}$), by the Euler class. In trivial case, null Euler class, we have that the fiber $\mathbb{S}^1$ represent one non trivial real homological class in $B\times \mathbb{S}^1$.

Question:

In the non trivial case (non zero Euler class), is the fiber homologous to zero in the total space (real homology)?

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Not necessarily. For example, for circle bundles over $S^2$, Euler class 2 corresponds to the circle bundle with total space $\mathbb R P^3$ with fiber representing the generator of $1$-dimensional homology of the total space.

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