When Are Two Coxeter Orbifolds Diffeomorphic?

One can define what it means for a compact manifold with corners to be a “contractible manifold with contractible faces.” Two combinatorially equivalent, contractible manifolds with contractible faces are diffeomorphic if and only if their 4-dimensional faces are diffeomorphic. It follows that two simple convex polytopes are combinatorially equivalent if and only if they are diffeomorphic as manifolds with corners. On the other hand, by a result of Akbulut, for each n ≥ 4, there are smooth, contractible n-manifolds with contractible faces which are combinatorially equivalent but not diffeomorphic. Applications are given to rigidity questions for reflection groups and smooth torus actions. AMS classification numbers. Primary: 57R18, 57R55, 20F55, 20F65 Secondary: 57S15, 57R91