What is a generic (genetic/geometric) map? (In the study of manifolds)

Question

At 29:30 in his lecture on Youtube, Mikhail Gromov talks about how one only gets a manifold from the zero set of the equation $f(0)=0$ if the map $f$ is "generic" (or genetic or geometric -- I mostly but not entirely understand his accent).

It is a dirty word, because it is extremely convenient, but you don't know what you get -- but on the other hand this is the major mechanism generating manifolds, by genericity(?)

He then goes on to discuss how this concept raises foundational issues, saying at 31:37 that if we do not allow such functions to exist then the continuum hypothesis is true, and that if we do allow such functions to exist, then the continuum hypothesis is false.

What concept is he referring to? It sounds extremely important.

The best I could find is generic point which sounds vaguely similar. That or perhaps he is mistranslating a Russian term and means regular point? Because I am aware of how one can use the implicit function theorem to take the inverse images of a regular point and create manifolds.

Also he talks in between about singularities being "rare" but coming with additional structure when they do occur, which sounds like Morse theory to me. He also talks about a result from Riemannian geometry which holds at least for dimensions 1-7 and was proved in part by Jim Simons.

Answer 1

Let $F$ be some space. A subset $C\subset F$ is generic if it contains a countable intersection of open and dense subsets. If $F$ is a Baire space, it follows that $C$ is dense itself. The intuition is that a generic set is in some sense "almost all" of the space. This generalizes the idea of a full measure set outside of measure theory. Often generic sets of interest are open themselves, but they need not be.

Some examples (some were given in the comments):

• The set of irrational numbers is generic as a subset of $\mathbb{R}$
• The set of polynomials of degree $n$ with $n$ distinct roots is generic
• The set of regular values of a smooth function $f:M\rightarrow \mathbb{R}$, with $M$ a smooth manifold, is generic (Sard's theorem). The set of critical points is not in general!
• The set of Morse functions on a given smooth manifold is generic (if the manifold is compact, it is open)
• The set of invertible matrices is generic

Given a smooth function $f:M\rightarrow \mathbb{R}$ the set of regular values is generic. This implies, that for "most" values $x\in \mathbb{R}$ the set $f^{-1}(x)$ is a smooth manifold. Indeed we apply the implicit function theorem here to conclude that $f^{-1}(x)$ is a smooth manifold, but the "most" does not come from the implicit function theorem. This is one of the easiest mechanisms to construct smooth manifolds.