For a closed connected -manifold smoothly embedded in , with and , its complement hasand zero reduced homology for , where negative-index groups are zero. The Thom isomorphism theorem for the normal bundle and the pair's exact sequence give this formula. The top ambient fundamental class maps isomorphically to the fundamental class of , canceling the exceptional top term.
A compact smooth -manifold with nonempty boundary embedded in is a proper compact locally contractible subset. Alexander duality therefore expresses the reduced homology of its complement by the reduced cohomology of the abstract manifold. Thus these groups do not depend on its particular smooth embedding.
Articles by others on the same topic
Alexander duality is a fundamental theorem in algebraic topology, specifically in the study of topological spaces and their homological properties. Named after mathematician James W. Alexander, the duality provides a relationship between the topology of a space and the topology of its complement. In its most basic form, Alexander duality applies to a locally finite CW complex, particularly when considering a subcomplex (or a subset) of a sphere.