In the following article
https://hal.archives-ouvertes.fr/hal-00148349/document
at the end of the proof of theorem 6.1 (p. 167), the author writes that he can conclude thanks to familiar arguments which I guess I'm not familiar with because I can't really see why $\tilde{u}$ induces an isomorphism. The conclusion I made (which is obvious) is that $\tilde{u}$ and $\tilde{v}$ have the same behavior on the homology but I don't understand what can help us conclude, knowing that $(S, \phi)$ is a resolution.
Anyway, if someone knows the answer, please guide me.