Antipodal quotient

The projective plane arises as a quotient space of the sphere, the required group being C,-. It is obtained by identifying antipodal points of the spherical surface in other words, it is the antipodal quotient of the sphere (see Section 1.2.2). P2 is the simplest compact non-orientable surface in the sense that it can be obtained from the sphere by adding just one cross-cap. 

In this terminology, our definition of projective fullerenes amounts to selection of cell-complex projective-planar 3-valent maps with only 5- and 6-gonal feces. As noted above, P5 — 6 for these maps. Thus, the Petersen graph is die smallest projective fullerene. In general, the projective fullerenes are exactly the antipodal quotients of the centrally symmetric spherical fullerenes. 

The real projective space RP can be represented as a CW complex with one cell in each dimension from 0 to n. This cell structure is the Z2-quotient, with respect to the antipodal map, of the cell structure on the sphere S , which we described in (l)(b) above. 

---