TITLE: Does the isomorphic of the fundamental groups imply the existence of a mapping inducing an isomorphism?
QUESTION [7 upvotes]: A pair of continuous mappings $f \colon X \to Y$ and $g \colon Y \to X$ is called $\pi_1$-equivalence if they induce mutually inverse isomorphisms of fundamental groups. Spaces are called $\pi_1$-equivalent if there is $π_1$-equivalence between them.
Let $X, Y$ be CW-complexes

Is it true that if $f \colon X \to Y$ induces an isomorphism of fundamental groups, then $X$ and $Y$ are $π_1$-equivalent?
Is it true that if $\pi_1(X)$ is isomorphic to $\pi_1(Y)$, then $X$ and $Y$ are $\pi_1$-equivalent?

(added later)

Is it true that if $\pi_1(X)$ is isomorphic to $\pi_1(Y)$, then there is of a mapping $f \colon X \to Y$  inducing an isomorphism or there is of a mapping $g \colon Y \to X$  inducing an isomorphism?

REPLY [5 votes]: As a consequence of Van Kampen's Lemma, in the special case where $X,Y$ are finite 2-dimensional CW-complexes then the answer is yes to all 3.