For a compact oriented -manifold with boundary, cap product with its relative fundamental class gives
Reverse a handle decomposition of a compact -dimensional manifold. Absolute -handles become relative -handles based on the boundary. The absolute boundary matrices count attaching sphere-belt sphere intersections; the corresponding relative matrices are their transposes over . The absolute chain complex is thus the complementary-degree dual of the relative chain complex. The universal coefficient theorem for cohomology proves Poincare-Lefschetz duality without an orientation hypothesis.
Articles by others on the same topic
Lefschetz duality is a powerful result in algebraic topology that relates the homology of a manifold and its dual in a certain sense. More specifically, it applies to compact oriented manifolds and provides a relationship between their topological features.