What are the exact critera for a CW-complex being a polytope?

Question

Everybody talks about the fact that polyhedra are special CW complexes, and some of the higher dimensional abstract polytopes are too, but nobody tells the exact criteria for a CW complex being a polytope (or I am clumsy a bit). What are these? Please give exact references for this (not only booknames, but pagenumbers too, if possible)!

Answer 1

Broadly speaking, a CW complex captures the structure of some polytope. How one defines a polytope is then pretty much the answer to which CW complexes are polytopes. For example if your definition of a polytope is general enough to include a 2-sphere with two inscribed edges, two vertices and two hemispherical faces (the digonal dihedron) as a polytope then yes, that CW complex is a polytope, but if your definition demands say three edges to any face of a polytope then no it is not.

Abstract polytopes and CW complexes are both classes of incidence complex. In abstract terms, for an incidence complex to qualify as a polytope:

• each element must be unique (i.e. the same edge cannot appear twice)

• the incidences must satisfy a "diamond" or "dyadic" condition (such that just two faces of a polyhedron meet at an edge and so on)

• it must be properly connected (so it cannot be divided into a compound of two structures, in the way that say the stella octangular is).

I seem to recall that all of these conditions also apply to CW complexes, but I am not sure. Note that there are also polytopes, such as projective polytopes, which are not CW complexes.

The standard reference book is McMullen & Schulte; Abstract Regular Polytopes. A useful short summary with slightly different terminology is provided by Norman Johnson in Geometries and Transformations, Chapter 11: Finite Symmetry Groups, pp. 223-226.