# Euler characteristic of homology theory of one object divides that of another

Suppose we have a homology theory such that the associated Euler characteristic of one object divides that of another. What can we infer from this?

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What sorts of properties are you hoping to infer? In the case of compact, closed surfaces, and singular homology, the Euler characteristic of the genus $g$ surface $\Sigma_{g}$ is $2(1-g)$. In particular, the Euler characteristic of the sphere $S^{2}=\Sigma_{0}$ divides the Euler characteristic of every $\Sigma_{g}$. As such, it would seem to be hopeless to infer anything - presumably you'd like to relate the topology/geometry of $\Sigma_{0}$ to $\Sigma_{g}$, for each $g$, or something else? –  George Melvin Feb 23 at 1:19