Deformations of smooth functions on $2$-torus

Abstract

Let $f$ be a Morse function on a smooth compact surface $M$ and ${\mathcal{S}}^{\prime }(f)$ be a group of $f$-preserving diffeomorphisms of $M$ which are isotopic to the identity map. Let also $G(f)$ be a group of automorphisms of the graph of $f$ induced by elements from ${\mathcal{S}}^{\prime }(f)$, and ${\Delta }^{\prime }$ be a subgroup of ${\mathcal{S}}^{\prime }(f)$ of diffeomorphisms which trivially act on the graph of $f$ and are isotopic to the identity map. The group ${\pi }_0{\mathcal{S}}^{\prime }(f)$ can be viewed as an analogue of a mapping class group for $f$-preserved diffeomorphisms of $M$. Groups ${\pi }_0{\Delta }^{\prime }(f)$ and $G(f)$ can be viewed as groups which encode `combinatorially trivial' and `combinatorially nontrivial' counterparts of ${\pi }_0{\mathcal{S}}^{\prime }(f)$ respectively. In the paper we compute groups ${\pi }_0{\mathcal{S}}^{\prime }(f)$, $G(f)$, and ${\pi }_0{\Delta }^{\prime }(f)$ for Morse functions on $2$-torus $T^2$.