I am now reading the article of M.F.Atiyah "Convexity and commuting hamiltonians" and I can't understand lemma 2.1. which says that if $\varphi \colon M \to \mathbb R$ is a Morse-Bott function without critical manifolds of index $1$ or $n-1$ then for each $c \in \mathbb R$ $\varphi^{-1}(c)$ is connected. Could someone help me with the proof or give a good reference where I can find it?
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1$\begingroup$ Is the intuition clear? The singularities of an index $1$ or $n-1$ critical set are the only places where level sets can become disconnected (near other-index singularities, the level sets near zero should have higher homotopy groups change but should remain connected). Maybe try using the Morse lemma near singularities and studying the resulting conics? Also note only need to prove for index $\leq n/2$. $\endgroup$– NealCommented Oct 3, 2016 at 15:18
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1$\begingroup$ There are only two Morse indices which allow changes in the number of components. It might help to think of passing through an index $i$ critical point as attaching a $D^i \times D^{n-i}$ along the $(\partial D^i) \times D^{n-i}$ and see what that operation does on the boundary. $\endgroup$– Ryan BudneyCommented Oct 3, 2016 at 17:19
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$\begingroup$ Check page 125 of these notes www3.nd.edu/~lnicolae/Morse2nd.pdf $\endgroup$– Liviu NicolaescuCommented Oct 3, 2016 at 19:50
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