# Morse theory and Euler characteristics

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).

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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.

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Ah that's a good point. Assuming constant signature on the components, can you say anything (even without Bott non-degeneracy in the normal direction). I'll try to think about it... –  Sam Lewallen Jan 4 '10 at 15:47