Corollary 1

If $C$ is split and $H(C)$ projective (resp. if $C'$ is split and $H_n(C')$ injective for each $n$), then the canonical homomorphism

is bijective.

Suppose for example $C'$ split and $H(C')$ injective for each $n$, the case where $C$ is split and $H(C)$ projective being proved in an analogous manner. By X, p. 35, def. 6, there exists a homotopy equivalence $u':C'\to H(C')$ ; by Proposition 3, $\mathrm{Homgr}_A(1,u')$ is a homotopy equivalence from $\mathrm{Homgr}_A(C,C')$ onto $\mathrm{Homgr}_A(C,H(C'))$ ; since

and that $\mathrm{Homgr}_A(1,H(u'))$ and $H(\mathrm{Homgr}_A(1,u'))$ are bijective, it is enough for us to prove that $\lambda(C, H(C'))$ is bijective, which brings us back to the case where $C'$ is injective and has zero differential.

The canonical exact sequences (X, p. 28)

(III)

(IV)

give exact sequences (X, p. 83, Proposition 2, a))

Since $d_C = i \circ \delta \circ p$, the differential $D$ of $\mathrm{Homgr}_A(C, C')$ is given by $D_n = (-1)^{n+1} P_n \circ \Delta_n \circ l_n$; we then have

whence an isomorphism $\varphi : H(\mathrm{Homgr}_A(C, C')) \to \mathrm{Homgr}_A(H(C), C')$ such that, if $a \in \mathrm{Homgr}_A(C/B(C), C')$, the image of the class of $P(a)$ under $\varphi$ is $J(a)$; we immediately verify that $\varphi$ coincides with the canonical homomorphism $\lambda$.