Stiefel-whitney Homology Classes of Quasi-regular Cell Complexes

A quasi-regular cell complex is defined and shown to have a reasonable barycentric subdivision. In this setting, Whitney's theorem that the ^-skeleton of the barycentric subdivision of a triangulated n-manifold is dual to the (n /c)th Stiefel-Whitney cohomology class is proven, and applied to projective spaces, lens spaces and surfaces. 1. QR complexes. A (finite) cell structure on a space X is defined (see, e.g., [Brown, p. 124]) to be a (finite) family of maps {a: E"°-> X) called cells so that (Ï)X= U0{a(int(£*))}. (ii) a\int(E"') is a homeomorphism onto its image. (iii) o(dE*) C U^<%(/x(£^)). (We will deal with finite complexes throughout for simplicity, although everything holds in the locally finite context.) The cell structure will be called quasi-regular (QR) if the following condition holds: for each cell a: E"° —> X, there is a cell structure (necessarily unique) on dE"° so that for each cell a in 8£n°, a ° « is a cell in X; note that this boundary structure must also be QR. Such a structure will be called a QR complex and X will denote the space X with this additional structure. The most familiar example of a QR structure is the usual cell structure on RP", denoted RP", with one cell in each dimension. Also, any regular cell complex is QR. The product of two QR structures is defined as usual by taking for cells in the product X X Y, products {a X r: EnX E"-> X X Y) of cells a and t in X and Y respectively, a structure which is easily checked to be QR. Barycentric subdivision can be defined inductively in analogy with the simplicial case as follows: for each nCT-cell a, cone over the cells in the subdivision of the associated QR structure on dE"° and consider the set of cells so obtained: a QR complex with only zero cells is its own subdivision. It is not hard to see that the subdivision of a QR complex A' is a QR complex, denoted X', and is in fact a pseudo-triangulation in the sense of [Hilton and Wiley], so that a second subdivision yields a triangulation. Incidence numbers can also be defined for a QR complex: if a and t are two cells in X, then [t, o] is the number of cells to in the QR structure on dE"' corresponding to t so that t ° w = a. In the barycentric subdivision, the origins of the original Received by the editors September 23, 1975 and, in revised form, May 17, 1976. AMS (MOS) subject classifications (1970). Primary 57C99; Secondary 57D20.