Is every group homomorphism the induced homomorphism of a continuous function?

Question

If $f : X \to Y$ is a continuous function and we fix base points $x_0 \in X$ and $y_0 = f(x_0)\in Y$, then $f$ induces a group homomorphism $$f_*: \pi_1(X, x_0) \to \pi_1(Y, y_0)$$ between the fundamental groups of $(X, x_0)$ and $(Y,y_0)$. I was just wondering if there is some kind of reciprocal property, that is:

If $\psi : \pi_1(X,x_0) \to \pi_1(Y,y_0)$ is a homomorphism, is it true that $\psi = f_*$ for some continuous function $f: X \to Y$?

Answer 1

Not in general, there are obstructions.

A discussion about this interesting problem is contained in MathOverflow question 166153.

Answer 2

The question in the title of your post and the question in the body are different. If you start with spaces $X$ and $Y$ and a homomorphism from $\pi_1(X,x_0)$ to $\pi_1(Y, y_0)$, then there are obstructions, as Francesco says. And this is probably what you had in mind.

However, it might be worth pointing out that if you start with an abstract homomorphism $\phi:G\to H$, then you can construct spaces $X$ and $Y$ with $\pi_1(X,x_0)=G$ and $\pi_1(Y,y_0)=H$, and a continuous function $f:X\to Y$ such that $f_*=\phi$. Namely, you can take X and Y to be Eilenberg Maclane spaces for G and H, respectively. See Hatcher page 90 for details.