Foliated circle bundles whose Euler class is torsion

Let $$X$$ be a closed manifold. By a foliated circle bundle $$E \rightarrow X$$ we mean a circle bundle over $$X$$ with total space $$E$$ and structure group $$Diff^+(S^1)$$, and a codimension one foliation of $$E$$ that is transverse to each circle fiber. Miyoshi states in Remark 2 of `On foliated circle bundles over closed orientable 3-manifolds', Comm. Math. Helv. , 1997 the following as a fact:

• For any torsion class $$e \in H^2(X; \mathbb{Z})$$ of any dimensional closed manifold $$X$$, there exists a $$C^\infty$$ foliated circle bundle over $$X$$ whose Euler class is equal to $$e$$. [i.e. the foliation is transversely $$C^\infty$$]

Could anyone give me a sketch of a proof, or point me to a reference please? Inspired by the above remark, a follow-up question is the following:

• Let $$X$$ and $$e$$ be as above. Is it true that if $$a_1 \in H^2(X; \mathbb{Z})$$ is realized as the Euler class of a foliated circle bundle over $$X$$, then $$a_2:=a_1 + e$$ is also realized?

Update: Igor Belegradek kindly mentioned a reference for the first part: Miyoshi, A remark on torsion Euler classes of circle bundles, Tokyo Journal of Mathematics, 2001.

He also added a reference in the comments which shows that a principal circle bundle has a flat connection if and only if its Euler class is a torsion. This answers the second question positively if we add the adjective principal to the circle bundle. See e.g. Theorem 3 of Milnor, One the existence of a connection with curvature zero, Comm. Math. Helv. 1958.

Theorem (Milnor): Let $$K$$ be a finite complex. The $$SO(n)$$-bundle over $$K$$ induced by any homomorphism $$\pi_1(K) \rightarrow SO(n)$$ has trivial Euler class with rational coefficients.

I don't know the answer to the second question for circle bundles which are not necessarily principal.

• There are two points: 1) any class in $H^2(X;\mathbb Z)$ is the Euler class of a (unique) oriented principal circle bundle over $X$, and 2) if a principal bundle has torsion Euler class, then it has a flat connection. Having a flat connection is equivalent to being foliated. For some references see my answer mathoverflow.net/questions/104974/…. Mar 29, 2020 at 13:56
• I added a reference in the edit. Mar 29, 2020 at 14:02
• Thanks, Igor. The reference you mentioned is great (a paper of Miyoshi from 2001 whose content is the proof of the above remark). Do you have any guess for the answer to the second question? Mar 29, 2020 at 14:09
• @IgorBelegradek I think the second reference is studying principal circle bundles [so the group of the bundle is S^1] whereas my question is about smooth circle bundles [so the group of the bundle is Diff(S^1)]. The if and only if statement is not true for circle bundles which are not necessarily principal circle bundles; I'm sure you are aware since you referred to MW inequality in your answer. Mar 29, 2020 at 14:23
• You might want to clarify the question then. It is not enough to say "circle bundle". You need to specify the structure group. Mar 29, 2020 at 14:26