Lefschetz duality

Remarks

The relative fundamental class is the class whose restriction to $H_n(M, M \setminus \set{x})$ is the local orientation at each interior point $x$. The proof uses a collar of the boundary to compare $M$ with its interior. It then applies the duality theorem behind Poincaré duality to the interior, a noncompact manifold without boundary. When $\partial M = \emptyset$, both isomorphisms reduce to Poincaré duality.

Example: for the disc $M = D^n$ with $\partial M = S^{n-1}$, $H^0(D^n) \cong \Z$ and $H_n(D^n, S^{n-1}) \cong \Z$. Also $H^n(D^n, S^{n-1}) \cong \Z$ matches $H_0(D^n) \cong \Z$, and all other groups vanish on both sides.

Going deeper

For a student

Lefschetz duality is the form of Poincaré duality used for compact manifolds with boundary, such as knot exteriors and cobordisms. Take a compact oriented $(2m+1)$-manifold $M$ and a field $F$. Lefschetz duality and the long exact sequence of $(M, \partial M)$ show that the kernel of $H_m(\partial M; F) \to H_m(M; F)$ has half the dimension of $H_m(\partial M; F)$. It is the basic tool in the study of manifolds with boundary and of bordism.