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

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    $\begingroup$ What is $\pi^1 (X)$? Is it $\pi_1(X)$? $\endgroup$ Oct 9 '13 at 20:51
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If $M$ is a one-dimensional Peano continuum and $N$ is a one-dimensional 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 non-trivial. The one-dimensional case can be used to show one-dimensional Peano continua with isomorphic fundamental groups are homotopy equivalent.

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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 2-skeleton 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 $2\leq k\leq dim(M)-1$.

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    $\begingroup$ Higher dimensional spheres are not contractible. You mean "simply connected". $\endgroup$ Oct 9 '13 at 23:08
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    $\begingroup$ What he means is that all higher dimensional spheres in $N$ are contractible. $\endgroup$ Oct 9 '13 at 23:27
  • $\begingroup$ Does that rewording fix the confusion? $\endgroup$
    – Ian Agol
    Oct 10 '13 at 0:12

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