# Evaluating the Euler class of a circle bundle on fibers

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 2-cycle on which the Euler class evaluates to 1 ? In other words, do we always have : $$\langle eu(\pi), H_2(M; \mathbb{Z}) \rangle = \mathbb{Z}$$ ?

• 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.
– mme
Nov 24, 2020 at 15:02
• (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.)
– mme
Nov 24, 2020 at 16:33
• 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. Nov 24, 2020 at 18:20
• @Mike Miller, in case this interests you : mathoverflow.net/questions/377365/… Nov 24, 2020 at 20:52