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Let $f,g:M \rightarrow N$ two smooth maps between smooth manifolds that are smoothly homotopic by $F$. Suppose also that $f$ and $g$ are transverse to a submanifold $A$ of $N$. I know that transverse maps are dense in the weak and strong topologies for the space of maps $C^{\infty}(M \times I, N)$ so its always possible to approximate the map $F$ by a map transverse to $A$. I also wish to find a transverse approximation to $F$ that is still a homotopy between $f$ and $g$, anyone knows if this is possible?

Vincent L.
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    Yes this is possible a proof can be found in differential topology by guillemin and pollack. – omar Jan 17 '14 at 16:57
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    Guillemin and Pollack only consider the $C^0$ topology on the space of smooth maps. You might also look at the Transversality Theorem in Hirsch's Differential Topology for consideration of $C^k$ topologies. – Ted Shifrin Jan 17 '14 at 17:27

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