Non-existence of nonorientable regular embeddings of n-dimensional cubes

By a regular embedding of a graph K into a surface we mean a two-cell embedding of K into a compact connected surface with the automorphism group acting regularly on flags. Regular embeddings of the n  -dimensional cubes QnQn into orientable surfaces exist for any positive integer n  . In contrast to this, we prove the nonexistence of nonorientable regular embeddings of QnQn for n>2n>2.