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If $M$ is a smooth manifold, then any framed submanifold $N$ is the preimage $f^{-1}(y) $ for a smooth sphere-valued map $f$ transversal to $y$, with the framing of the normal bundle induced by $f$.

My question is: which framed submanifolds are induced by $\mathbb{R}^n$-valued maps? More specifically, what is the condition on a framed submanifold $N$ to be the preimage $f^{-1}(0)$ for some $f$ transversal to $0$? For example, a nontrivial circle on the 2-torus is not the zero set of an $\mathbb{R}$-valued map, but two circles (with compatible framing) are. Also, note that $4$ points on a circle with framing $(+,+,-,-)$ are not the zero set of an $\mathbb{R}$-valued transversal map, although they are framed null-cobordant. Thanks for any hint!

Peter Franek
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  • 4 points on a circle are the zero set of a Morse function. Bend the circle into a horseshoe shape. – Cheerful Parsnip May 18 '14 at 06:50
  • By the "+" in the framing, I mean that the normal vector goes in the clock-wise direction, so the function changes from negative to positive (if you go clockwise). This can not happen, on a circle, once and again. But if you forget the framing, then any even number of points on a circle is the zero set. However, how to generalize this? (I think that framed null-cobordant is a necessary condition but not sufficient, as this example shows) – Peter Franek May 18 '14 at 06:56
  • Oh sorry, I missed that your signs were ordered. I agree this can't be realized. – Cheerful Parsnip May 18 '14 at 06:59
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    FYI: I asked this question also on mathoverflow: http://mathoverflow.net/questions/167415/which-submanifolds-are-zero-sets-of-mathbbrn-valued-maps – Peter Franek Jun 29 '14 at 11:35

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