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We study the decomposition of a stratum $\mathcal H(\kappa)$ of abeliandifferentials into regions of differentials that share a common$L^\infty$-Delaunay triangulation. In particular, we classify the infinitelymany adjacencies between these isodelaunay regions, a phenomenon whoseobservation is attributed to Filip in work of Frankel. This classificationallows us to construct a finite simplicial complex with the same homotopy typeas $\mathcal H(\kappa)$, and we outline a method for its computation. We alsorequire a stronger equivariant version of the traditional Nerve Lemma thancurrently exists in the literature, which we prove.

The $L^\infty$-isodelaunay decomposition of strata of abelian differentials