chapter  26
15 Pages

Poincaré Duality

WithMarvin J. Greenberg, John R. Harper

Throughout this section X will denote an n-dimensional manifold. If X is R-orientable, we choose once and for all an R-orientation of X; otherwise, we take R = Z/2 and take the unique Z/2-orientation (22.12). For each compact K ⊂ X, restricting the R-orientation (regarded as a section of the R-orientation sheaf) to K and using the isomorphism Hn (X, X – K) → ΓK gives a fundamental class ζK ∈ Hn (X, X – K) (22.21 and 22.24). This notation will be used throughout this section.