My question is as follows. Is there known general results on typical singularities (critical points) for smooth maps from $m$-dimensional torus $\mathbb{T}^{m}=\mathbb{R}^{m}/\mathbb{Z}^{m}$ to $\mathbb{R}^{n}$, $n \geq m$ (though the case $n<m$ is interesting too).
I'm new to this one and I've just looked for answers in "Arnold V. I., Gusein-Zade S. M., Varchenko A. N., Singularities of Differentiable Maps: Volume I: the Classification of Critical Points Caustics and Wave Fronts, Birkhäuser, Berlin (1985)". This book is out of my current math level, but here are a few things that I think I've understood:
- For $n > 2m$: regular maps (with no critical points) are typical.
- For $n=m=2$: classical result of H. Whitney: 2 types of singularities and stable maps are typical.
- For $m=2,n=3$: classical result of H. Whitney: 3 (?) types of singularities and stable maps are typical.
And there are so called transversality theorems, which I can't figure out for now, but I feel they can help me maybe.
I will be grateful for any help.
