Real elements in the mapping class group of \(T^{2}\) (Q712214)

Let \(Y\) be a compact connected oriented smooth \(3\)-manifold, and \(F\) an oriented smooth \(2\)-manifold. An \(F\)-fibration \(\pi : Y \to S^1\) is called weakly real if there is an orientation preserving diffeomorphism \(H : Y \to Y\) which sends fibers into fibers reversing their orientations. If \(H^2 = id\), then \(\pi\) is called real. In this paper, this structure is investigated because of its relationship with the real structures of Lefschetz fibrations. A diffeomorphism \(f : F \to F\) is called weakly real if there is an orientation reversing diffeomorphism \(c : F \to F\) such that \(f^{-1} = c \circ f \circ c^{-1}\). If \(c^2 = id\), then \(f\) is called real. It is shown that an \(F\)-fibration is real (resp. weakly real) if and only if its monodromy is real (resp. weakly real). In this paper, the condition of \(f\) to be real is investigated in the case where \(F = T^2\). The mapping class group of \(T^2\) is identified with \(SL(2, \mathbb{Z})\) via its action on the first homology group. The elements of \(SL(2, \mathbb{Z})\) are classified into three types: if \(| tr (A) | <2\) then \(A\) is elliptic, if \(| tr(A) | = 2\) then \(A\) is parabolic, and if \(| tr(A) | >2\) then \(A\) is hyperbolic. By a linear fraction transformation, \(SL(2, \mathbb{Z})\) acts on the upper half plane. The action of a hyperbolic matrix \(A\) on the upper half plane is encoded by a cutting period-cycle \([a_1, \ldots, a_{2n}]_A\). The main result of this paper is: (1) all elliptic and parabolic matrices in \(SL(2, \mathbb{Z})\) are real, (2) a hyperbolic matrix \(A \in SL(2, \mathbb{Z})\) is real if and only if its cutting period-cycle \([a_1, \ldots, a_{2n}]_A\) is odd-bipalindromic, (3) a matrix in \(SL(2, \mathbb{Z})\) is real if and only if it is weakly real.