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


1. Is it true that if $f \colon X \to Y$ induces an isomorphism of fundamental groups, then $X$ and $Y$ are $π\_1$-equivalent?
2. Is it true that if $\pi\_1(X)$ is isomorphic to $\pi\_1(Y)$, then $X$ and $Y$ are $\pi\_1$-equivalent?


(added later)


3. 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?



Does the isomorphic of the fundamental groups imply the existence of a mapping inducing an isomorphism?
No and no. For an explicit counterexample to 1. (which is also a counterexample to 2.) take the map $\mathbb{R}P^2\to \mathbb{R}P^{\infty}$.


