Suppose that $X$ is a simplicial complex, and $f:X \rightarrow S^k$ a continuous map to a sphere. Is $f$ always homotopic to a simplicial map to the boundary of a $(n+1)$-simplex, $\partial \Delta^k$?


The simplicial approximation theorem implies that there is a barycentric subdivision of $X$, $X'$, and a simplicial map $g:X'\rightarrow \partial \Delta^k$ that is homotopic to $f$. The question is whether we can drop the barycentric subdivision in this particular case.


EDIT: this question originally asked "Does a nonzero cohomology imply existence of a simplicial map to a sphere?" Indeed, as Bertram Arnold pointed out, the complex projective plane has nonzero homology in degree 2, but has no non-null-homotopic maps to the 2-sphere.



Is every map from a simplicial complex to a sphere homotopic to a simplicial map to the boundary of a k-simplex?
Take $X = \partial \Delta^3$. Then homotopy classes of maps from $|X|$ to $S^2$ correspond to elements of $\pi\_2(S^2)\cong \mathbb{Z}$. But there are only finitely many maps of simplicial complexes $\partial \Delta^3 \to \partial \Delta^3$. So not all of them can be represented without further subdivision.


