On the realization space of the cube

Given a d-dimensional polytope P, its realization space consists of all d-polytopes that have the same combinatorial type as P. For instance, consider the realization space of the d-simplex: since the d-simplex is projectively unique (i.e., any d-simplex can be transformed into any other d-simplex by a projective transformation) the realization space of the d-simplex is connected (and in fact, contractible). As a consequence, any two simplicial d-polytopes, after applying an appropriate projective transformation to one of them, can be glued along a facet to produce a convex polytope. This realizes the connected sum operation for simplicial polytopes geometrically. 

We consider the realization space of the d-cube. The d-cube is not projectively unique for , and we cannot realize the connected sum operation geometrically for cubical d-polytopes, . We enlarge the set of transformations, and show that any two realizations of the d-cube are connected by a finite sequence of transformations from this larger set, and that the realization space of the d-cube is contractible. Furthermore, we use this sequence to define an analog of the connected sum operation for cubical d-polytopes. I will mention generalizations to other families of polytopes, and an application of the cubical connected sum.

Based on joint work with Karim Adiprasito and Eran Nevo.