of ${\partial}M$ via the projection $\pi \colon M {\longrightarrow}X$ defines a subset $\pi({\partial}M)$, which can be identified with $Y$, and $\pi({\partial}M)$ forms the unique singular stratum of the stratified space $X$. (See, for instance [@ALMPI] or [@DLR2011], among many others, for a discussion of stratified spaces.) In the case $Z \simeq \mathbb{S}^f$ a sphere, $X$ is homeomorphic to a manifold. In general, given $p \in
\pi({\partial}M)$, there is a basis of open sets at $p$ consisting of *canonical neighborhoods* homeomorphic to $$\label{eq:canon-neighb}
  V \times C_\delta(Z),$$ where $V$ is an open subset of $Y$ containing $p$ and $$C_\delta(Z) = [0, \delta) \times Z / {\left\{ 0 \right\} } \times Z.$$

Differential form bundles and the Hodge-Laplacian
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We will consider differential forms and vector fields which are of approximately unit size with respect to Riemannian metrics of the type in . These are the incomplete cusp edge forms, which are sections of the incomplete cusp edge form bundle, ${{}^{\textrm{ice}}\Lambda^{*}}(M)$, whose smooth sections are generated locally over the base by the forms $$\label{eq:ice-forms}
  dx, \qquad dy_{i}\quad (i=1,\ldots,b=\dim Y), \qquad x^{k} dz_{\alpha}\quad (\alpha=1,\ldots,f=\dim Z).$$ Locally over the base, we have the isomorphism $$\label{eq:ice-decomp}
  {{}^{\textrm{ice}}\Lambda^{*}}(M) = \Lambda^{*}(Y) \wedge x^{k \operatorname{\mathbf{N}}}\Lambda^{*}(Z)  \oplus  dx \wedge
  \Lambda^{*}(Y) 