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?
Homotopy groups from the dual graph
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algebraic-topology