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Let $M$ be a compact smooth manifold and $N$ be a compact smooth submanifold of $M$. The usual transversality theorem claims that for a generic diffeomorphism $f$ of $M$, the submanifolds $N$ and $f(N)$ are transverse.

I am interested in another point of view. The diffeomorphism $f$ is now fixed and I wonder now if one can perturb smoothly the submanifold $N$ so that $N$ and $f(N)$ are transverse. Of course it requires some condition on $f$ as trivially it cannot be done for the identity map. Assume maybe first $f$ has no periodic points in $M$.

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  • $\begingroup$ Welcome to MO! "trivially it can not be done for the identity map"? I would rather say trivially it can be done with the identity map, you can make a submanifold transverse to itself with arbitrarily small deformations. $\endgroup$ – Arnaud Mortier Jan 24 '18 at 14:01
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    $\begingroup$ @Arnaud I think you misread the question. The OP asking if for a submanifold $N$, does there exist a deformation $N_d$ of $N$ with $f(N_d)$ transverse to $N_d$. This is clearly never exists for $f$ the identity unless dim N= dim M. $\endgroup$ – PVAL Jan 24 '18 at 14:19
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    $\begingroup$ I am interesting in the transversality of $\tilde{N}$ and $f(\tilde{N})$ where $\tilde{N}$ is the perturbation of $N$. Not $f(\tilde{N})$ and $N$... $\endgroup$ – user119986 Jan 24 '18 at 14:22
  • $\begingroup$ In fact the good notion should be a notion of genericity between two maps $X\rightarrow Y$ that implies in particular that the set where the two maps agree is discrete, and the required property should be for $f$ here to be generic with respect to Id$_M$. $\endgroup$ – Arnaud Mortier Jan 24 '18 at 20:05

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