Suppose we have a space M with a real-valued, differentiable function F on M. Under what conditions on F will the Euler characteristic of M be expressed as a (signed) sum of Euler characteristics of components of the critical set for F? Can we relax the Morse-Bott requirement? What if the critical set isn't smooth... can we ever say anything?
Thanks! (I've also just asked a longer companion question on similar things).
You could have a smooth function $f : \Bbb R \to \Bbb R$ whose critical point set is a Cantor set (minima) and the centres of the complementary intervals (local maxima) -- let $f$ be some suitable smoothing of the distance function from the Cantor set (or you could use the smooth Urysohn lemma to construct the function), say. The two sets (local max / local min) don't have the same cardinalities so you've got no hope of a formula.
I suppose the most direct case where it should fail for the critical point set a manifold, is when the Hessian is non-degenerate in the normal directions but having non-constant signature over an individual critical component. Off the top of my head I don't have an example but the proof (Bott's proof) fundamentally breaks down in this situation so this is where I would look first.
(1) look up "Stratified Morse Theory" by Goresky-MacPherson; they show that in the restricted case of a stratified space (e.g., algebraic variety), there is a well-defined Morse-Euler formula for functions that are more general than Bott-Morse. (2) if you fix an o-minimal structure (see "Tame Topology and O-Minimal Structures" by Van den Dries), then a tame real-valued function will satisfy a Morse-Euler formula without any Morse-type restrictions on the function; without any restrictions on the underlying space (other than tameness). (3) if you're not dealing with stratified or tame spaces, you are probably out of luck without some other restrictions. even infinitely differentiable functions can have nasty critical sets. note that real-analyticity leads to stratified sets --- you can get somewhere in that case.
In symplectic geometry, one often uses the norm-square of the moment map as a Morse function. However, in general it is slightly more degenerate than a Morse-Bott function. The norm-square of the moment map satisfies a weaker condition called minimal degeneracy, defined by Kirwan in Cohomology of Quotients in Symplectic and Algebraic Geometry. Kirwan shows that one may obtain Morse inequalities (and much more) from such functions. It is easy to check that all Morse-Bott functions are minimally degenerate in this sense, so Kirwan's approach gives a useful generalization of Morse-Bott theory.
