# Stratification of the space of maps transverse to another given one

If $$M,N,S$$ are manifolds ($$M$$ and $$S$$ compact) and $$g:S\to N$$ is a smooth map, then if we endow $$C^\infty(M,N)$$ with the strong topology we get that $$\pitchfork(M,N;g):=\{f\in C^\infty(M,N)\,;\,f\pitchfork g\}$$ is dense and open in $$C^\infty(M,N)$$ (see Hirsch, Differential Topology, and Kosinski, Differential Manifolds). This indicates that $$\pitchfork(M,N;g)$$ would fit to be the $$0$$-stratum of some stratification on $$C^\infty(M,N)$$.

I would be interested to know if someone already established such stratification, with a particular interest in the description of the $$1$$-stratum he or she came with. I am also very interested if the article discusses how a generic path $$f_t\in C^\infty(M,N)^{[0,1]}$$ behaves with respect to this stratification. Thanks for your help!