Let $M$ and $N$ be two connected, smooth (possibly non-compact), aspherical manifolds (that is, $\pi_k(-)=1$ for $k\geq 2$) of dimensions $n$ and $n-r$ respectively. Let $f:M\to N$ be a smooth map inducing a surjective homomorphism $f_*:\pi_1(M)\to \pi_1(N)$. What can we say about the kernel of $f_*$? Is the cohomological dimension of the kernel $r$?