Which homomorphism $\pi_1(M) \to \pi_1(N)$ can be realized as induced homomorphism of a continuous map $f:M\to N$?

2$\begingroup$ What is $\pi^1 (X)$? Is it $\pi_1(X)$? $\endgroup$– Johannes EbertOct 9 '13 at 20:51
If $M$ is a onedimensional Peano continuum and $N$ is a onedimensional metric space, then the homomorphism is induced by a continuous map $f:M\to N$ up to a basepoint change. This is a result of Katsuya Eda.
The same holds if $M$ and $N$ are planar Peano continua (Greg Conner and Curtis Kent).
These cases do not assume local contractability and are quite nontrivial. The onedimensional case can be used to show onedimensional Peano continua with isomorphic fundamental groups are homotopy equivalent.
I don't think there is a simple general answer to this question (I'm assuming here that $M$ and $N$ are reasonable topological spaces like CW complexes). If $N$ is a $K(\pi,1)$, then this is possible. One may map the 2skeleton of $M$ to $N$ using the homomorphism, and then map the higher skeleta in using the fact that maps of higher dimensional spheres into $N$ are homotopically trivial. This also works if the higher homotopy groups $\pi_k(N)$ vanish for $2\leq k\leq dim(M)1$.

1$\begingroup$ Higher dimensional spheres are not contractible. You mean "simply connected". $\endgroup$ Oct 9 '13 at 23:08

1$\begingroup$ What he means is that all higher dimensional spheres in $N$ are contractible. $\endgroup$ Oct 9 '13 at 23:27
