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I think that looking at surfaces one can see that there is no nice relation between both sides. Let $\Sigma_g = \#^g S^1 \times S^1$ be the oriented closed surface of genus $g$. Then one can show, using simplical volumes aka the Gromov norm (see http://www.map.mpim-bonn.mpg.de/Simplicial_volume), which satisfies $||\Sigma_g|| = 4g-4$, that for $g,h >1$, we have $$D(\Sigma_g,\Sigma_h) = \left\{0,\pm 1, \dots, \pm \left\lfloor \frac{g-1}{h-1}\right\rfloor\right\}.$$ In particular we have $D(\Sigma_g,\Sigma_h) = \{0\}$ for $g < h$. I leave it to you to conclude form this that there is no nice relation (just choose $g$ and $h$ in the right way...).