## Tokyo Journal of Mathematics

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

Shigeaki MIYOSHI

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 19 October 2009

http://projecteuclid.org/euclid.tjm/1255958322

Digital Object Identifier
doi:10.3836/tjm/1255958322

Mathematical Reviews number (MathSciNet)
MR1844428

Zentralblatt MATH identifier
1011.57008

#### Citation

MIYOSHI, Shigeaki. A Remark on Torsion Euler Classes of Circle Bundles. Tokyo J. of Math. 24 (2001), no. 1, 189--194. doi:10.3836/tjm/1255958322. http://projecteuclid.org/euclid.tjm/1255958322.