# Generalizations of Sard-Smale Theorem

Sard-Smale theorem holds for Fredholm maps $$f:M\rightarrow B$$ between separable Banach manifolds $$M,N$$. There are some constrains relating the Fredholm index $$\operatorname{ind}(f)$$ of $$f$$ to its differentiablity class. More precisely, we need to require $$f\in C^{r}$$, where $$r>\max{(\operatorname{ind}(f),0)}$$.

I would like to know what are the generalizations of this theorem into two directions:

1. assuming weaker regularity on $$f$$;
2. assuming weaker structure on the spaces $$M,N$$.

For instance, I know that the result is valid for bounded Fréchet manifolds in the case where $$f$$ is Lipchitz (see here). The result is also valid for some domains with empty interior (see here).

I would also like to know if the above (or others) generalizations are maximal, in the sense that there are counterexamples in the case of weaker requirements.

Any help is welcome.

Edit: I posted the question on MathOverflow.

• Even in the finite dimensional case there are counterexamples with weaker regularity. I think these are due to Whitney. I recall hearing somewhere that there are more precise versions that are sharp (?), but I don't recall from where. – Thomas Rot Apr 14 '20 at 9:02
• Thank you for the comment. – Math-Phys-Cat Group Apr 14 '20 at 12:48
• I'm new in these forums, so that I do not know exactly how the things work. Should I move my question to mathover flow? Thanks. – Math-Phys-Cat Group Apr 15 '20 at 15:45