How to imagine/prove that all of the following pictures are a portion of a torus? The obvious way is counting number of genus, but it isn't easy for me to count it. any suggestion? (image source: MO-Renato G. Bettiol and Cassidy Curtis website)
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8$\begingroup$ Nice pictures . $\endgroup$– user732848Commented Aug 11, 2020 at 8:52
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1$\begingroup$ What do you know about Morse theory? $\endgroup$– Noel LundströmCommented Aug 11, 2020 at 11:36
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$\begingroup$ Somewhat. I know the Morse index theorem (number of critical points) and its relation to Euler characteristics. $\endgroup$– C.F.GCommented Aug 11, 2020 at 11:52
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2$\begingroup$ It's a little hard from these pictures but you should be able to count the critical points (of type 0, 1 and 2 respectively) and compute the Euler characteristic $\chi$, we also know that $\chi = 2 - 2g$ where $g$ is the genus. I also assume that all of the shapes depicted are symmetric by rotation of 120 degrees? $\endgroup$– Noel LundströmCommented Aug 11, 2020 at 12:34
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1 Answer
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The comments are misinformed: a monkey saddle is not a Morse singularity. You cannot find the Euler characteristic by counting critical points, since they are not all nondegenerate!
Instead you must use the following variation of the Morse Lemma.
Consider the sequence of groups $H_*(M_t)$, where $M_t = f^{-1}(-\infty, t]$. This sequence changes precisely for critical values $t$. So we just need to see what happens at the Monkey saddle (let's say that happens at time $t=1$). For $M_{.999}$, we have a manifold diffeomorphic to a disc, as it has exactly one nondegenerate critical point.
Now look at the picture in IV. What happens as we go from the bottom to the top? We've taken $M_{.999} \times [0,1]$ and added a "tripod" on the top --- a space that looks like a thickened letter $Y$ --- by attaching the three boundary arcs $(\text{3 points}) \times I$ to $M_{.999} \times \{1\}$. You can see by the Mayer-Vietoris sequence that the result has $H_*(M_{1.001}) = \Bbb Z$ in degree 0 and $\Bbb Z^2$ in degree 3. Another way to think about this: attching this "tripod" is functionally equivalent to attaching two handles, which correspond to two nondegenerate index 1 critical points.
Anyway, all that's left is to attach the cap, which only adds something to top degree homology. What we get out of this is that the homology of this surface matches up with the torus, and so it is a torus.
