Let $f:M\rightarrow\mathbb{R}$ be a smooth function, where $M$ is a closed manifold. I have the feeling that the Lusternik-Schnirelmann category of $M$ is an estimate from below of the number of components of the critical points set of $f$, rather than only the number of critical points of $f$, which is, of course, a stronger result. Am I wrong? Any references are welcome.
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Doesn't the height function on a torus laying down on its side give a counter-exsmple? There are two circles of critical points, but $\mathrm{cat}(T^2) =3$.
Added: A constant function on a connected non-contractible manifold gives an even easier example.
answered Aug 20, 2016 at 5:53
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$\begingroup$ Thank you for the answer, I see that this is not the right question. Anyway, I have in mind some examples, when the function has some number of components of the critical set and it seems stable by perturbations. So, I wondered what invariant would be responsible for that. $\endgroup$ Commented Aug 20, 2016 at 20:06
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$\begingroup$ You might be interested in Morse-Bott homology. As well as the original paper of Bott on periodicity, this is discussed in an article of Austin and Braam. I can dig out the precise references later if you like. $\endgroup$ Commented Aug 21, 2016 at 5:30