  We study the low energy resolvent of the Hodge Laplacian on a manifold
equipped with a fibred boundary metric. We determine the precise asymptotic
behavior of the resolvent as a fibred boundary (aka $\phi$-) pseudodifferential
operator when the resolvent parameter tends to zero. This generalizes previous
work by Guillarmou and Sher who considered asymptotically conic metrics, which
correspond to the special case when the fibres are points. The new feature in
the case of non-trivial fibres is that the resolvent has different asymptotic
behavior on the subspace of forms that are fibrewise harmonic and on its
orthogonal complement. To deal with this, we introduce an appropriate 'split'
pseudodifferential calculus, building on and extending work by Grieser and
Hunsicker. Our work sets the basis for the discussion of spectral invariants on
$\phi$-manifolds.
