### A Remark on Torsion Euler Classes of Circle Bundles

Shigeaki MIYOSHI
Source: Tokyo J. of Math. Volume 24, Number 1 (2001), 189-194.

#### Abstract

We show that any torsion class $e\in H^2(M;Z)$ of any closed manifold $M$ is realized as the Euler class of a smoothly foliated orientable circle bundle over $M$. In the case where $M$ is a 3-manifold, we construct the homomorphism $\pi_1(M)\rightarrow SO(2)\subset \text{Diff}_{+}^{\infty}(S^{1})$ explicitly whose Euler class is the given torsion class.

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