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Abstract: For a Weyl polytope $P$, there is an associated real toric variety $X_P^{\mathbb{R}}$. The quotient of $X_P^{\mathbb{R}}$ by the Weyl group action is obtained by gluing copies of cubes together. In this lecture, we will see the sufficient condition for the glued subspace of the cube to be contractible, and hence the corresponding gluing operation is a homotopy equivalence.