Construct $S^3$ as the double of a convex Euclidean polyhedron. This can be seen as a cell decomposition of $S^3$, with $V$ 0-cells, $E$ 1-cells and $F$ 2-cells (the number of vertices, edges and faces of our polyhedron respectively); and 2 3-cells. Now, we can construct a graph inside $S^3$ as follows: start with 2 nodes, one at the centre of each copy of the polyhedron, and for each face of the polyhedron join them with an edge passing once through that face. Call this embedded graph $\Gamma$.

Now for the question: is it true that $S^3$ minus the 1-skeleton in the construction above is homotopy equivalent to $\Gamma$?


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