A pair of continuous mappings $f \colon X \to Y$ and $g \colon X \to Y$ 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 $\pi_1(Χ)$ is isomorphic to $\pi_1(Y)$, then $X$ and $Y$ are $\pi_1$-equivalent?