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I have a closed (compact without boundary) manifold M and a compact Lie group G that acts on it. I want to understand the topology of $M/G$, at least compute its singular homology groups. The action isn't free, so $M/G$ may not be a manifold. Because of this, I can't apply classical Morse Theory to M/G. In classical Morse Theory we would construct a Morse function $\overline{f}\colon M/G \to \mathbb{R}$, for which every critical point has a non-degenerate Hessian. Instead, I can construct a smooth function $f\colon M \to \mathbb{R}$ which is invariant by the action, so it induces a continuous function $\overline{f}\colon M/G \to \mathbb{R}$.

Are there some properties that $f\colon M \to \mathbb{R}$ should satisfy, so that we get something equivalent to a Morse function for $M/G$? My guess is to use Morse-Bott functions, which are a generalization of Morse functions. A Morse-Bott function is one for which the critical points are submanifolds, and for each critical point the Hessian is non-degenerate in the normal directions of the corresponding critical submanifold. I would require $f\colon M \to \mathbb{R}$ to be a Morse-Bott function, such that the Hessian is non-degenerate in the normal directions to the orbit of the critical point. Notice that the orbits are submanifolds, since the action is proper, which always holds for compact Lie groups. I expect this condition to force the critical points of $f$ to be isolated orbits, which should project to isolated points in $M/G$. Does this gives me something equivalent to a Morse function for M/G? I'm struggling to find a reference for this.

Edit: I've found the reference MORSE THEORY FOR FLOWS IN PRESENCE OF A SYMMETRY GROUP (https://apps.dtic.mil/sti/tr/pdf/ADA144661.pdf). Not sure yet if it solves my problem.

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    $\begingroup$ This is about equivariant cohomology (basically where you attach a copy of $B G_x$ to $M/G$ at each non-free point $x$, at least up to homotopy) as opposed to just ordinary cohomology of the quotient, but I imagine it could be helpful at least in looking for refs: mathoverflow.net/questions/171512/… $\endgroup$ Commented Dec 1 at 17:37
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    $\begingroup$ Whow, that's what I call a username! On a more serious note, you can always turn a non-free action into a free action by attaching, at each non-free point $x\in M$, the classifying space $BG_{x}$ of the stabilizer group, and study Morse functions on the resulting space. I am, however, not entirely sure what your objective is. $\endgroup$ Commented Dec 1 at 20:31
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    $\begingroup$ If you had a $G$-CW decomposition of $X$ (inductively pasting $G$-spaces of the form $G/H \times D^n$), quotienting would give a CW decomposition of $X/G$ which you could understand explicitly. The problem is that Morse theory does not give $G$-CW decompositions, it gives decompositions into representation spheres. Said another way, at $x \in X$ with stabilizer $\Gamma \subset G$, it is typically not the case that $T^-_x X$ has trivial $\Gamma$-representation. If you can ensure this is true, the computation from there should be easy. $\endgroup$ Commented Dec 1 at 23:37
  • $\begingroup$ Have a look at austin and braam webhomes.maths.ed.ac.uk/~v1ranick/papers/braamaus.pdf $\endgroup$ Commented Dec 2 at 1:26
  • $\begingroup$ What kinds of things are you hoping to say about the quotient object? If your question is literally as general as how you've stated it, you might not have many tools available. If you are okay saying fairly simple things about the quotient, there may be useful tools. $\endgroup$ Commented Dec 2 at 2:02

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