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For a connected oriented topological surface of finite type with negative Euler characteristic, every orientation-preserving mapping class has a periodic representative, preserves an essential multicurve, or has a pseudo-Anosov map representative. In the usual disjoint classification, the reducible case means a nonperiodic class preserving such a multicurve. The assertion concerns representatives up to isotopy, rather than requiring an arbitrary homeomorphism in a periodic class to have finite order.

For a torus with one puncture, the action on first homology identifies its mapping class group with $S{L}_2(\mathbb{Z})$. For the corresponding matrix $M$, the periodic classes have $|\mathrm{tr}M|<2$ or $M=\pm I$. Nonperiodic reducible classes have $|\mathrm{tr}M|=2$ and $M\ne \pm I$; pseudo-Anosov map classes have $|\mathrm{tr}M|>2$. For example, $\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$ is a Dehn twist, while $I$ is periodic despite having the same trace.^[ref]

A puncture is distinguished from a boundary circle fixed pointwise: boundary twists give additional mapping classes invisible to the displayed $S{L}_2(\mathbb{Z})$ action. For the closed torus, the hyperbolic case has an Anosov homeomorphism representative.