Take $n = 2$,$r = 0$ and $M = \mathbb R^2 - \{0,1\}$ and $N = \mathbb R^2 - \{0\}$ and $M \to N$ to be the natural inclusion.

Then on fundamental groups we have the surjection $F_2 \to \mathbb Z$ taking one generator to $1$ and the other to $0$.  The kernel is a nontrivial subgroup of $F_2$, hence free and of cohomlogical dimension $1 \neq 0$.