Take on the spheres $\mathbb{S}^n$ and $\mathbb{S}^m$ some "easy" simplicial structures $\Sigma^n$ and $\Sigma^m$ (e.g. as surfaces of the corresponding simplices). Then by simplicial approximation any homotopyclass of maps $\pi_n(\mathbb{S}^m)\ni f:\mathbb{S}^n\rightarrow \mathbb{S}^m$ can be realized by some simplicial map $\mathfrak{f}:BSD^k(\Sigma^n)\rightarrow\Sigma^m$ for some finite $k$, where $BSD^k$ denotes the $k$-fold barycentric subdivision. Now Serre proved, that (allmost) all the $\pi_n(\mathbb{S}^m)$ are finite, so in particular for $n,m$ fix, there is a $k$ such that we can realize any map $f\in\pi_n(\mathbb{S}^m)$.
Now the actual question is the following: Are there any known bounds on this $k$ or maybe some direct bounds on the number of simplices in a simplicial decomposition of $\mathbb{S}^n$ needed to realize $\pi_n({S}^m)$ and can these be used to determine information about the higher homotopy groups of the spheres?
Edit: I duplicated this question on MathOverflow: https://mathoverflow.net/questions/308400/how-many-cells-are-needed-in-a-simplicial-structure-of-mathbbsn-to-induce