Why are the level sets connected for a Hamiltonian $S^1$-action?

Question

In these lecture notes by Weimin Chen, in Corollary 1.8, it is stated that

Let $H : M → \mathbb{R}$ be a moment map of a Hamiltonian $S^1$-action on a compact, connected manifold M. Then each level surface $H^{−1} (\lambda)$ is connected.

The author states, in the paragraph above, that this follows by "a standard argument in Morse theory" from the fact that $H$ is Morse-Bott with even index at all critical submanifolds.

My question is what is this standard argument / why does this follow?

Answer 1

I found a proof in

McDuff, Dusa; Salamon, Dietmar, Introduction to symplectic topology, Oxford Mathematical Monographs. New York, NY: Oxford University Press (ISBN 0-19-850451-9/pbk). x, 486 p. (1998). ZBL1066.53137.

in Lemma 5.51. While the statement of their lemma is slightly different and weaker, their proof shows the given Corollary 1.8.

The key feature of the moment map is that it is a Morse-Bott function whose critical submanifolds have index and coindex even (which follows from an equivariant Darboux theorem as explained by Weimin Chen in the linked notes). For convenience, I give an outline of the given proof below

Proof outline:

Let $f : M \to \mathbb{R}$ be Morse-Bott with even index and coindex at all critical submanifolds. Let the critical values be $c_0 < c_1 < \ldots < c_N$.

1. Show that the critical submanifold $C_0$ of index zero is connected. This follows because its stable manifold is a complement of codimension $\ge 2$ submanifolds (the other stable manifolds). It follows that $C_0 = f^{-1}(c_0)$.
2. Show that for $c_0 < c < c_1$, $f^{-1}(c)$ is connected: take any two points in $f^{-1}(c)$, flow them down to $C_0$, connected these points by a path in $C_0$, move this path slightly away from $C_0$, and then flow it up to $f^{-1}(c)$ to connect the given points.
3. More generally, show by induction that $f^{-1}(c)$ is connected for $c_j < c < c_{j+1}$: take any two points, connect them via a path in $f^{-1}(c)$ to points in the stable manifold of $C_0$, flow these down to below the critical submanifold $f^{-1}(c_j)$, use induction hypothesis to connect the points in $f^{-1}(c_j - \epsilon)$ by a path, perturb the path so it is transverse to the unstable manifolds, and hence so it's disjoint from them, and flow it back up to $f^{-1}(c)$.
4. Notice that, for $1 \le j < N$, the map $f^{-1}(c_j + \epsilon) \to f^{-1}(c_j)$ given by flowing back is a surjective (continuous) map, hence (by general topological reasons), $f^{-1}(c_j + \epsilon)$ being connected implies $f^{-1}(c_j)$ is connected also.
5. Finally, $f^{-1}(c_N)$ is connected by repeating the argument in 1. for $-f$.