= Nielsen–Thurston classification
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= Nielsen–Thurston classification theorem
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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 $SL_2(\mathbb Z)$. For the corresponding matrix $M$, the periodic classes have $|\operatorname{tr}M|<2$ \b[or] $M=\pm I$. Nonperiodic reducible classes have $|\operatorname{tr}M|=2$ and $M\ne\pm I$; <pseudo-Anosov map> classes have $|\operatorname{tr}M|>2$. For example, $\begin{pmatrix}1&1\\0&1\end{pmatrix}$ is a <Dehn twist>, while $I$ is periodic despite having the same trace.<A primer on mapping class groups (Version 5.0) by Benson Farb and Dan Margalit>[Section 13.1, torus classification, pages 388–390]{ref}
A puncture is distinguished from a boundary circle fixed pointwise: boundary twists give additional mapping classes invisible to the displayed $SL_2(\mathbb Z)$ action. For the closed <torus>, the hyperbolic case has an <Anosov homeomorphism> representative.
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