\v{C}ech complexes for finite sets

Abstract

Let $\mathcal{F}^{m}=\{A\subset[m]\}$, endowed with the symmetric difference metric and the Cardinality Reverse-Lexicographic order. That way, we compute the homotopy type $\operatorname{\v{C}ech}\left(\mathcal{F}_{\preceq a}^m; \frac{r}{2}=1\right)$ by relating it to consecutively adding vertices of $\mathbb{I}^m$, the unit hypercube of $m$th dimension, to point $0$. Further, for higher $r$, we propose a conjecture regarding contractibility of a subcomplex implying a formula for the exact number of wedges of $\mathbb{S}^ 3$'s and $\mathbb{S}^ 4$'s in the $\operatorname{\v{C}ech}$ complex.