Partitioning the triangles of the cross polytope into surfaces

Abstract

We present a constructive proof that there exists a decomposition of the 2-skeleton of the k-dimensional cross polytope $β^k$ into closed surfaces of genus $g\le 1$, each with a transitive automorphism group given by the vertex transitive ${\mathbb{Z}}_{2k}$-action on $β^k$. 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 Möbius strips.

13 pages, 1 figure. Minor update. Journal-ref: Beitr. Algebra Geom. / Contributions to Algebra and Geometry, 53(2):473-486, 2012