  We consider pairs of few-body Ising models where each spin enters a bounded
number of interaction terms (bonds), such that each model can be obtained from
the dual of the other after freezing $k$ spins on large-degree sites. Such a
pair of Ising models can be interpreted as a two-chain complex with $k$ being
the rank of the first homology group. Our focus is on the case where $k$ is
extensive, that is, scales linearly with the number of bonds $n$. Flipping any
of these additional spins introduces a homologically non-trivial defect
(generalized domain wall). In the presence of bond disorder, we prove the
existence of a low-temperature weak-disorder region where additional summation
over the defects have no effect on the free energy density $f(T)$ in the
thermodynamical limit, and of a high-temperature region where in the
ferromagnetic case an extensive homological defect does not affect $f(T)$. We
also discuss the convergence of the high- and low-temperature series for the
free energy density, prove the analyticity of limiting $f(T)$ at high and low
temperatures, and construct inequalities for the critical point(s) where
analyticity is lost. As an application, we prove multiplicity of the
conventionally defined critical points for Ising models on all $\{f,d\}$
tilings of the hyperbolic plane, where $df/(d+f)>2$. Namely, for these infinite
graphs, we show that critical temperatures with free and wired boundary
conditions differ, $T_c^{(\mathrm{f})}<T_c^{(\mathrm{w})}$.
