Let $M$ be a $n$ manifold such that $\pi_k(M)$ is non trivial. What can we expect about the regularity of a representant $f:S^k\rightarrow M$ of a non-trivial cycle? For example, if $M$ is a manifold of class $p$, is it possible to find a $C^p$ function? I heard that these kind of problems were studied by the Russian school some decades ago, so I am mostly asking for a reference, but I would also like to know if there existed an answer to this specific question.
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asked Jul 13, 2015 at 22:21
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1$\begingroup$ For the $C^\infty$ and $C^\omega$ cases this has been answered in mathoverflow.net/questions/203627/… $\endgroup$– ThiKuCommented Jul 14, 2015 at 4:01
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$\begingroup$ Thank you very much ThiKu, but if $p$ is a finite positive integer, and the manifold is no more than $C^p$, the question still needs an answer. But I will check in details Whitney's result; it would not be too surprising if his proof could be adaptable to lower regularity. Of course, if there is a direct reference, I would really appreciate the information. $\endgroup$– Paul-BenjaminCommented Jul 14, 2015 at 7:28
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1 Answer
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I've often heard it quoted as a Corollary of this paper that every homotopy class $\alpha: S^k\to M$ is represented by a map with only fold singularities (of type $\Sigma^{1,0}$):
Èliašberg, Ja. M. Singularities of folding type. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 34 1970 1110–1126.
answered Jul 14, 2015 at 6:16
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$\begingroup$ Thank you very much Mr. Grant, I will close the question if a get an answer about low regularity manifolds. $\endgroup$ Commented Jul 14, 2015 at 7:29
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1$\begingroup$ You're welcome @Paul-Benjamin. There's no need to close the question (you can accept an answer, but in my opinion people are often too quick to do this as it discourages other would-be answerers). $\endgroup$ Commented Jul 14, 2015 at 9:47
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$\begingroup$ Thank you for your understanding, it is what I thought too. $\endgroup$ Commented Jul 14, 2015 at 9:49