Orientations and Poincaré duality

For an n-dimensional manifold without boundary, the local group $H_n(M,M\setminus\{x\};\mathbb Z)$ is infinite cyclic. An orientation chooses compatible generators as x moves through coordinate neighborhoods. On a connected compact manifold without boundary, a global orientation produces a fundamental class $[M]\in H_n(M;\mathbb Z)$ restricting to every chosen local generator. Compactness and absence of boundary are encoded in the word closed here; it does not merely mean closed as a subset of some ambient Euclidean space.

A connected closed nonorientable manifold has no such integer fundamental class and has zero top integer homology. With F₂ coefficients the sign obstruction disappears and every manifold is orientable over that field. Orientation is therefore a coefficient-sensitive notion. On a compact manifold with boundary, the appropriate top class is relative to the boundary. On a noncompact manifold, a global class may require locally finite chains rather than an ordinary finite singular chain. These distinctions are structural, not adjustments made after an incorrect formula fails.

For a connected closed oriented smooth n-manifold, cap product with its integer fundamental class gives an isomorphism $H^k(M;\mathbb Z)\to H_{n-k}(M;\mathbb Z)$ in every degree. This includes torsion; it is stronger than a symmetry of real Betti numbers. A proof combines local calculations with excision and compatibility of cap products, often assembling local orientation data through Mayer–Vietoris. Merely matching the ranks of two groups would not prove the integral isomorphism or identify its natural construction.

Over an orientation coefficient field, compactness gives finite-dimensional groups, and evaluation of complementary cup products on the fundamental class is a nondegenerate pairing. Over Z, ordinary integer-valued evaluation cannot be perfect on torsion: a homomorphism from a finite group to Z is zero. One therefore separates the free-part intersection form from additional torsion linking information. The integral duality isomorphism itself remains valid; what fails is the shortcut that interprets every torsion class through an ordinary integer-valued bilinear pairing.

For a compact oriented manifold with boundary, Poincaré–Lefschetz duality pairs ordinary cohomology with relative complementary homology, and relative cohomology with ordinary complementary homology. For an oriented noncompact manifold without boundary, compactly supported cohomology pairs with ordinary complementary homology. Nonorientable manifolds can use suitable orientation local coefficients, or F₂ coefficients for an untwisted version. These are different statements with different groups; the closed-manifold formula should not be applied first and repaired by ignoring the discrepancy.

Reversing the global orientation negates the fundamental class and hence changes the sign of top-degree evaluations, without changing whether duality is an isomorphism. For a closed oriented surface, degree-one products form a skew-symmetric pairing over characteristic zero, reflecting signed intersections of loops. This gives a practical consistency check on calculations of cup products and map degree. Still, a visually crossing pair of curves must be assigned orientations and transverse-intersection conventions before its signs can be compared with an algebraic formula.