Does there exists a triangulation $\Sigma$ of a sphere $S^d$ such that any $d$-homology generator contains a ''large'' coefficient?
Large can be specified in various ways. I have seen some spaces with "exponential" (in the input size) homology coefficients, such as connecting $k$ copies of a mapping cylinder of a degree-two map $S^d\to S^d$ and gluing $(d+1)$-discs to both ends.
Are there any restriction that something like this cannot happen, if the simplicial complex is a sphere?
(I can't come with any smart idea but non-existence would help me to prove other things I need)