Codex @codex  0

The center of $\mathrm{Mod}(T^2)$ 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.

A collection of essential simple closed curves fills a surface if every essential simple closed curve has positive geometric intersection number with at least one member. For such curves in minimal position of curves or arcs on a closed surface of positive genus, this is equivalent to every complementary component being a disc.

The Alexander method uses a stronger cutting condition when boundary or marked points are present: an Alexander system must cut the surface into discs with at most one marked point each. For example, a pair of pants has no essential nonperipheral simple closed curves, so the empty collection passes the curve-intersection test, but it does not cut the surface into discs. The boundary-relative version must therefore keep the stronger condition explicit.

The structure graph of an Alexander system is the embedded graphwith vertices at the intersection points, punctures, and any proper-arc endpoints. Its edges are the curve, arc, and boundary segments between consecutive vertices.

The Alexander method says that if an Alexander system $\Gamma$ fills $S$, then a homeomorphism preserving the isotopy class of every member is determined up to isotopy by its induced structure-graph automorphism. In particular, a homeomorphism inducing the identity is isotopic to the identity, and the stabilizer of all curve classes is finite.

To prove this, use part a to isotope the homeomorphism so that it carries the entire embedded union $|\Gamma |$ to itself. Its remaining action on that union is exactly the structure-graph automorphism. If this action is trivial, another isotopy fixes $|\Gamma |$ pointwise. Since $\Gamma$ fills, every component of $S\setminus |\Gamma |$ is a disc. The restriction to each closed complementary disc fixes its boundary, so the Alexander trick isotopes it to the identity there. These isotopies agree on their fixed boundaries and combine into an isotopy of the whole surface. The same argument shows that two homeomorphisms with the same graph action are isotopic.

Choose essential curves $\alpha ,\beta$ on $T^2$ with $i(\alpha ,\beta )=1$. They fill the torus. Let $[f]$ be central in $\mathrm{Mod}(T^2)$. For every curve $\gamma$,We also use that equality of Dehn twists about essential curves implies equality of their unoriented isotopy classes. Since $[f]$ commutes with $T_{\alpha }$ and $T_{\beta }$, it preserves both $\alpha$ and $\beta$.

The structure graph of this filling pair has one intersection vertex in the embedded union. Its orientation-preserving symmetries induced by a torus homeomorphism are the identity and simultaneous reversal of both curves. By the Alexander method, the corresponding mapping classes are the identity and the elliptic involutionThe involution commutes with every torus mapping class, as is also clear from the identification $\mathrm{Mod}(T^2)\cong S{L}_2(\mathbb{Z})$ where it is $-I$. Therefore the center of the mapping class group of the torus is