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A projective (or free) $\mathbb{Z}\pi_1$-module

Based on your comments to this answer, you seem to be in the situation where $\pi_2(Z)$ is a free $\mathbb{Z}\pi_1(X)$-module, and where the map $\pi_1(X)\to \pi_1(Y)$ is an iso. Under those assumptio …
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