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.

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