gn.general topology - Morse theory and Euler characteristics

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.