  The dual of a polyhedron is a polyhedron -- or in graph theoretical terms:
the dual of a 3-connected plane graph is a 3-connected plane graph.
Astonishingly, except for sufficiently large facewidth, not much is known about
the connectivity of the dual on higher surfaces. Are the duals of 3-connected
embedded graphs of higher genus 3-connected, too? If not: which connectivity
guarantees 3-connectedness of the dual? In this article, we give answers to
this and related questions. Among other things, we prove that there is no
connectivity that for every genus guarantees the 3-connectedness or
2-connectedness of the dual, and give upper bounds for the minimum genus for
which (with c>2) a c-connected embedded graphs with a dual that has a 1- or
2-cut can occur. For the torus, we determine exact values for the connectivity
needed to guarantee 3- respectively 2-connectivity of the dual. We prove that
already on the torus, we need 6-connectedness to guarantee 3-connectedness of
the dual and 4-connectedness to guarantee 2-connectedness of the dual.
  In the last section, we answer a related question by Plummer and Zha on
orientable embeddings of highly connected non-complete graphs.
