Suppose we have a polytope with dual graph $G^*$ (each facet is a vertex, and two facets are adjacent iff they share a codimension 1 polytope). Is there any way to compute homotopy groups of this polytope from its dual graph, or if there is a way to tell if a certain homotopy group is nontrivial?
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The dual graph is the 1-skeleton for the dual cellulation of the polytope. Therefore it can tell you $\pi_0$. The inclusion of the dual graph into the polytope also induces a surjection on $\pi_1$. So you get some information.