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I start with a longer discussion which will result in a precise version of the question. A am puzzled about an issue with the Quillen plus construction. I have seen outstanding experts being confused about this point. There are the following different ways of calling a map $f:X \to Y$ a homology equivalence:

3'. The homotopy fibres of $f$ are acyclic. 3'''. $f$ is can be identified with the Quillen plus construction.

What Segal and McDuff prove is that if the action is by weak homology equivalences, then $\phi$ is a weak homology equivalence. This is what is typically used to established the above results. To prove that 1,2,3 are strong homology equivalences, one can invoke an extra argument which is specific to each case.

I convinced myself that the whole argument goes through with strong homology equivalences (and the corresponding notion of "strong homology fibration"). Proposition 2 loc.cit. then has the assumption that $M$ acts on $M_{\infty}$ by strong homology equivalences (one needs the notion of homology equivalences one wants to prove in the end - which I find plausible).

This amounts, say in example 4, to prove that the stable stabilization map $B Out(F_{\infty}) \to B Out(F_{\infty})$ is a homology equivalence in the strong sense. For "weak homology equivalence", one invokes the usual homology stability theorem (Hatcher-Vogtmann-Wahl). But it seems that for the map being a strong homology equivalence, one needs a stronger homological stability result. I can imagine how the homological stability arguments can be modified to include abelian coefficient system, but that is not a satisfying solution.

I start with a longer discussion which will result in a precise version of the question. I am puzzled about an issue with the Quillen plus construction. I have seen outstanding experts being confused about this point. There are the following different ways of calling a map $f:X \to Y$ a homology equivalence:

3'. The homotopy fibres of $f$ are acyclic. 3'''. $f$ can be identified with the Quillen plus construction.

What Segal and McDuff prove is that if the action is by weak homology equivalences, then $\phi$ is a weak homology equivalence. This is what is typically used to establish the above results. To prove that 1,2,3 are strong homology equivalences, one can invoke an extra argument which is specific to each case.

I convinced myself that the whole argument goes through with strong homology equivalences (and the corresponding notion of "strong homology fibration"). Proposition 2 loc. cit. then has the assumption that $M$ acts on $M_{\infty}$ by strong homology equivalences (one needs the notion of homology equivalences one wants to prove in the end - which I find plausible).

This amounts, say in example 4, to prove that the stable stabilization map $B Out(F_{\infty}) \to B Out(F_{\infty})$ is a homology equivalence in the strong sense. For "weak homology equivalence", one invokes the usual homology stability theorem (Hatcher-Vogtmann-Wahl). But it seems that for the map to be a strong homology equivalence, one needs a stronger homological stability result. I can imagine how the homological stability arguments can be modified to include abelian coefficient system, but that is not a satisfying solution.

1. $f_\*:H_\*(X;\mathbb{Z}) \to H_*(Y;\mathbb{Z})$ is an isomorphism ("weak homology equivalence").
2. For each abelian system of local coefficients $A$ on $Y$ ($\pi_1 (Y)$ acts through an abelian group), the induced map $H_* (X;f^* A) \to H_* (Y;A)$ is an isomorphism ("strong homology equivalence").
3. For each system of local coefficients $A$ on $Y$, the induced map $H_* (X;f^* A) \to H_* (Y;A)$ is an isomorphism ("acyclic map").

1. $f_*:H_*(X;\mathbb{Z}) \to H_*(Y;\mathbb{Z})$ is an isomorphism ("weak homology equivalence").
2. For each abelian system of local coefficients $A$ on $Y$ ($\pi_1 (Y)$ acts through an abelian group), the induced map $H_* (X;f^* A) \to H_* (Y;A)$ is an isomorphism ("strong homology equivalence").
3. For each system of local coefficients $A$ on $Y$, the induced map $H_* (X;f^* A) \to H_* (Y;A)$ is an isomorphism ("acyclic map").

EDIT: Before, I included the statement ''3''. $f$ is weak homology equivalence, $\pi_1 (f)$ is epi and $ker(\pi_1 (f))$ is perfect.'' This is false (does not imply the other two conditions); in my answer to Spaces with same homotopy and homology groups that are not homotopy equivalent? I gave an example of a weak homology equivalence that is even an isomorphism on $\pi_1$, but whose homotopy fibre is not acyclic. END EDIT

EDIT: Before, I included the statement ''3''. $f$ is weak homology equivalence, $\pi_1 (f)$ is epi and $ker(\pi_1 (f))$ is perfect.'' This is false (does not imply the other two conditions); in my answer to Spaces with same homotopy and homology groups that are not homotopy equivalent? I gave an example of a weak homology equivalence that is even an isomorphism on $\pi_1$, but whose homotopy fibre is not acyclic. END EDIT