  We study various operator homological properties of the Fourier algebra
$A(G)$ of a locally compact group $G$. Establishing the converse of two results
of Ruan and Xu, we show that $A(G)$ is relatively operator 1-projective if and
only if $G$ is IN, and that $A(G)$ is relatively operator 1-flat if and only if
$G$ is inner amenable. We also exhibit the first known class of groups for
which $A(G)$ is not relatively operator $C$-flat for any $C\geq1$. As
applications of our techniques, we establish a hereditary property of inner
amenability, answer an open question of Lau and Paterson, and answer an open
question of Anantharaman--Delaroche on the equivalence of inner amenability and
Property (W). In the bimodule setting, we show that relative operator
1-biflatness of $A(G)$ is equivalent to the existence of a contractive
approximate indicator for the diagonal $G_\Delta$ in the Fourier--Stieltjes
algebra $B(G\times G)$, thereby establishing the converse to a result of
Aristov, Runde, and Spronk. We conjecture that relative $1$-biflatness of
$A(G)$ is equivalent to the existence of a quasi-central bounded approximate
identity in $L^1(G)$, that is, $G$ is QSIN, and verify the conjecture in many
special cases. We finish with an application to the operator homology of
$A_{cb}(G)$, giving examples of weakly amenable groups for which $A_{cb}(G)$ is
not operator amenable.
