Let $M^n$ and $\overline{M}^{n+1}$ be smooth manifolds, with $M$ compact. Suppose $M$ can be immersed in $\overline{M}$. Is it true that $M$ can be immersed in $\overline{M}$ in such a way that it always intersects itself transversely?
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2Yes, though I don't have a reference for you. – Nov 18 '16 at 23:35
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I strongly believe this is true, but I don't know a proof for it.. – Eduardo Longa Nov 18 '16 at 23:38
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@Mike: http://math.stackexchange.com/questions/1448843/reference-for-self-intersections-of-immersions is related, but definitely not the same thing. There, you recommend Hirsch. I don't have a copy - could this result be there? – Jason DeVito - on hiatus Nov 19 '16 at 00:49
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Yes, try Hirsch exercise 1 p83. – Nov 19 '16 at 00:50