  We present a constructive proof, that there exists a decomposition of the
2-skeleton of the k-dimensional cross polytope \beta^k into closed surfaces of
genus \leq 1, each with a transitive automorphism group given by the vertex
transitive Z_{2k}-action on \beta^k. Furthermore we show, that for each k
\equiv 1,5(6) the 2-skeleton of the (k-1)-simplex is a union of highly
symmetric tori and M\"obius strips.
