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 $r$?

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 $r$?

Edit: Thanks to user19232801 for answering my question. Now, I would like to correct/update it to what exactly I wanted. Assume that the cohomological dimensions of $\pi_1(M)$ and $\pi_1(N)$ are $n$ and $n-r$, respectively, $0<r<n$.