Deformations of smooth functions on $2$-torus

Let $f $ be a Morse function on a smooth compact surface $M$ and $\mathcal{S}'(f)$ 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 $\mathcal{S}'(f)$, and $\Delta'$ be a subgroup of $\mathcal{S}'(f)$ of diffeomorphisms which trivially act on the graph of $f$ and are isotopic to the identity map. The group $\pi_0\mathcal{S}'(f)$ can be viewed as an analogue of a mapping class group for $f$-preserved diffeomorphisms of $M$. Groups $\pi_0\Delta'(f)$ and $G(f)$ can be viewed as groups which encode `combinatorially trivial' and `combinatorially nontrivial' counterparts of $\pi_0\mathcal{S}'(f)$ respectively. In the paper we compute groups $\pi_0\mathcal{S}'(f)$, $G(f)$, and $\pi_0\Delta'(f)$ for Morse functions on $2$-torus $T^2$.

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