$\pi_1$-equivalence of CW-complexes Definition. We call $\pi_1$-equivalence the closure of a binary relation between topological spaces "there is a continuous mapping inducing an isomorphism of fundamental groups" to an equivalence relation.
Questions:

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*Are there two non-$\pi_1$-equivalent CW-complexes with isomorphic fundamental groups?
(here Achim Krause gave an example of two CW-complexes with isomorphic fundamental groups between which there is no continuous mapping inducing isomorphism, but are there any invariants that could impede the chain of such mappings?).

*If so, what are the $\pi_1$-equivalence invariants of CW-complexes?

It is clear that a CW-complex is $\pi_1$-equivalent to its
2-skeleton, so we can restrict ourselves to two-dimensional
CW-complexes.
 A: For any pair of 2-dimensional (connected) CW-complexes $X$ and $Y$, if $\pi_1(X)$ is isomorphic to $\pi_1(Y)$ then there exist maps going each way $X \mapsto Y$ and $Y \mapsto X$ that induce isomorphisms on $\pi_1$. Using what you have already written that the 2-skeleton is $\pi_1$-equivalent to the whole complex, it follows that the only invariant of $\pi_1$-equivalence is the fundamental group itself.
For the proof, first pick $0$-cells $p \in X$ and $q \in Y$ and an isomorphism $\phi : \pi_1(X,p) \to \pi_1(Y,q)$. Now define $f : X \to Y$ as follows. Pick a maximal subtree $T$ in the $1$-skeleton $X^{(1)}$, hence $T$ contains each $0$-cell. Define the restriction $f \mid T$ to be the constant with value $q$.
Next, orient and enumerate the edges $\{e_i\}_{i \in I}$ of $X^{(1)}-T$. Associated to each $e_i$ is an element $[\gamma_i] \in \pi_1(X,p)$ where $\gamma_i = \alpha_i e_i \beta_i$, $\alpha_i$ is the path in $T$ from $p$ to the initial $0$-cell of $e_i$, $\beta_i$ is the path in $T$ from the terminal $0$-cell of $e_i$ to $p$. So far the map is already defined on $\alpha_i$ and on $\beta_i$ to map those paths to the single point $q$, because $f$ is already defined on $T$ so as to take all of $T$ to $q$. Therefore $f$ is already defined on the edge $e_i$ so as to map the endpoints of $e_i$ to the single point $q$. Now extend $f$ over the rest of the edge $e_i$, defining $f \mid e_i$ to be any closed path in $Y$ based at $p$ which represents $\phi[\gamma_i]$.
It follows that, so far, $f$ has been defined on the closed path $\gamma_i = \alpha_i * e_i * \beta_i$ based at $p$ so as to take $[\gamma_i]$ to a closed path $f(\gamma_i)$, based at $q$, representing $\phi[\gamma_i]$.
Finally, for any $2$-cell $d$, its attaching map $\chi_d : S^1 \to X^1$ is homotopic to some path of the form $\gamma_{i_1} \cdots \gamma_{i_K}$, and then since $[\gamma_{i_1} \cdots \gamma_{i_K}]\in \pi_1(X,p)$ is the identity it follows that $\phi[\gamma_{i_1} \cdots \gamma_{i_K}] \in \pi_1(Y,q)$ is the identity and so $f(\gamma_{i_1} \cdots \gamma_{i_K})$ is homotopic to a constant in $Y$. From this it follows that $f \circ \chi_d$ is homotopic to a constant in $Y$. Using this homotopy we can therefore extend $f$ continuously over $d$.
