I am trying to understand what kind on information the Euler class provides about certain submanifolds of a given circle bundle. This might be completely obvious, but I don't see how to answer the following question : let $\pi^* : V \to M$ be a principal $S^1$bundle, and denote by $eu(\pi) \in H^2(M; \mathbb{Z})$ its Euler class. Is it true that there always exists a 2cycle on which the Euler class evaluates to 1 ? In other words, do we always have : $\langle eu(\pi), H_2(M; \mathbb{Z}) \rangle = \mathbb{Z}$ ?
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5$\begingroup$ Why would you expect this? It is not true. If $M$ is a surface, then $e \in H^2(M;\Bbb Z) \cong \Bbb Z$ is an integer, and the pairing with $H_2(M;\Bbb Z) \cong \Bbb Z$ is just multiplication. So your question is only true for exactly two $U(1)$bundles  the one with Euler class 1 and the one with Euler class 1. Perhaps even more to the point when $V = M \times S^1$ the Euler class is trivial, so the pairing you are asking about is zero. $\endgroup$– mmeNov 24, 2020 at 15:02

1$\begingroup$ (The first question was not intended to be harsh, though in retrospect unfortunately it sounds that way. I wanted to understand where you were coming from; perhaps there's something you wanted to use this to prove.) $\endgroup$– mmeNov 24, 2020 at 16:33

$\begingroup$ Thank you @Mike Miller. Of course, I didn't take it that way :). But you're right, the answer to my question was obviously negative. Indeed, this question was intended to clarify some concepts related to prequantization bundles. I will write another question in this matter, and will be happy to exchange with you about it. $\endgroup$– BrianTNov 24, 2020 at 18:20

$\begingroup$ @Mike Miller, in case this interests you : mathoverflow.net/questions/377365/… $\endgroup$– BrianTNov 24, 2020 at 20:52
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