Hmm. A quad-edge map in n-dimensions on a closed surface? I dunno.
Quad-edge data structures are used in three dimensional mappings. They provide what you seem to want. I do not have clue on generalizing.
The only reference I have is:
L J. Guibas & J Stolfi,
"Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams", ACM Transactions on Graphics, 4(2), 1985, 75-123
which is old. Sorry I can't provide something more substantive. Stolfi has libquad code here in C: