  We define a type of modulus $\operatorname{dMod}_p$ for Lipschitz surfaces
based on $L^p$-integrable measurable differential forms, generalizing the
vector modulus of Aikawa and Ohtsuka. We show that this modulus satisfies a
homological duality theorem, where for H\"older conjugate exponents $p, q \in
(1, \infty)$, every relative Lipschitz $k$-homology class $c$ has a unique dual
Lipschitz $(n-k)$-homology class $c'$ such that $\operatorname{dMod}_p^{1/p}(c)
\operatorname{dMod}_q^{1/q}(c') = 1$ and the Poincar\'e dual of $c$ maps $c'$
to 1. As $\operatorname{dMod}_p$ is larger than the classical surface modulus
$\operatorname{Mod}_p$, we immediately recover a more general version of the
estimate $\operatorname{Mod}_p^{1/p}(c) \operatorname{Mod}_q^{1/q}(c') \leq 1$,
which appears in works by Freedman and He and by Lohvansuu. Our theory is
formulated in the general setting of Lipschitz Riemannian manifolds, though our
results appear new in the smooth setting as well. We also provide a
characterization of closed and exact Sobolev forms on Lipschitz manifolds based
on integration over Lipschitz $k$-chains.
