  We study the decomposition of a stratum $\mathcal H(\kappa)$ of abelian
differentials into regions of differentials that share a common
$L^\infty$-Delaunay triangulation. In particular, we classify the infinitely
many adjacencies between these isodelaunay regions, a phenomenon whose
observation is attributed to Filip in work of Frankel. This classification
allows us to construct a finite simplicial complex with the same homotopy type
as $\mathcal H(\kappa)$, and we outline a method for its computation. We also
require a stronger equivariant version of the traditional Nerve Lemma than
currently exists in the literature, which we prove.
