Center of the mapping class group of the torus
= Center of the mapping class group of the torus
{title2=$Z(\operatorname{Mod}(T^2))=\{1,\iota\}$}
The center of $\operatorname{Mod}(T^2)$ has order two. A central class commutes with twists about two curves intersecting once, hence preserves both curve classes. The <Alexander method> reduces it to the identity or the elliptic involution $\iota:x\mapsto-x$, and both are central.