:= M / \{ p \sim q \mid p, q \in {\partial}M \mbox{ and } \pi(p) = \pi(q)\}.$$ In \[sec:hodge-theory\] we will prove the following.

\[thm:hodge\] For a cusp edge space $(M, g_{ice})$ whose link $Z$ satisfies the Witt condition , there is a natural isomorphism $$\label{eq:Hodge-isomorphism-general}
    \mathcal{H}^*_{L^2}(M, g_{ice}) \simeq I\!H_{{\overline{{\mathfrak{m}}}}}(X),$$ where $I\!H_{{\overline{{\mathfrak{m}}}}}$ is the middle perversity intersection cohomology of $X$. Furthermore, differential forms $\gamma \in \mathcal{H}^*_{L^2}(M, g_{ice})$ admit asymptotic expansions at the boundary of $M.$

Moreover, if $Z \simeq \mathbb{S}^{f}$, the sphere of dimension $f$, then $X$ is homeomorphic to a differentiable manifold and the isomorphism becomes $$\label{eq:Hodge-isomorphism}
    \mathcal{H}^*_{L^2}(M, g_{ice}) \simeq H^*_{dR}(X),$$ where the latter is the de Rham cohomology of $X$.

We recall the relevant facts about intersection cohomology, originally defined by Goresky and MacPherson in [@GM1980; @GM1983], in Section \[sec:hodge-theory\] below. The equivalence in will follow using the arguments from Hunsicker and Rochon’s recent work [@HR2012] on iterated fibered cusp edge metrics (which are *complete*, non-compact Riemannian manifolds). To elaborate on the asymptotic expansion for $L^2$-harmonic forms $\gamma$, we will show in Lemma \[thm:phg-kernel\] below that in fact $$\mathcal{H}^*_{L^2}(M, g_{ice}) = {\left\{   \alpha \in L^2(\Omega^p(M), g_{ice}) \mid
    \Delta^{g_{ice}} \alpha = 0  \right\} },$$ (that the former is 