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.



Regularity of maps in algebraic topology for manifolds
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.


