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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!

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