What is the intuition behind the following theorem of Reeb?
If a compact manifold admits a function with only two critical points which are non degenerate, it is homeomorphic to the sphere.
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4$\begingroup$ en.wikipedia.org/wiki/Morse_theory $\endgroup$– José Figueroa-O'FarrillCommented Aug 12, 2015 at 0:30
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2$\begingroup$ A Morse function on a smooth manifold $M$ gives rise to a handlebody decomposition of $M$, with an $i$-handle for each index $i$ critical point. In the case of Reeb's theorem, there is one index $0$ critical point (minimum) and one index $n$ critical point (maximum), and so $M$ is obtained by attaching an $n$-handle to a $0$-handle, i.e., gluing two $n$-balls by identifying their boundary $\mathbb{S}^{n-1}$'s. However, this is simply $\mathbb{S}^n$ (Alexander's trick). $\endgroup$– Amitesh DattaCommented Aug 12, 2015 at 1:13
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$\begingroup$ This question may be helpful: math.stackexchange.com/questions/1270099/… $\endgroup$– Dan RamrasCommented Aug 12, 2015 at 16:53
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