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.

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