Lefschetz duality

Lefschetz–Poincaré duality

An assertion about the duality between homology and cohomology, established by S. Lefschetz. More precisely, if $(X,A)$ is a pair of spaces such that $X\setminus A$ is an $n$-dimensional topological manifold, then for any Abelian group $G$ and any $i$ there is an isomorphism

$$H_i(X,A;G)\approx H_c^{n-i}(X\setminus A;G).$$

On the right-hand side one has cohomology with compact support. If the manifold $X\setminus A$ is non-orientable, one must, as usual, take cohomology with local coefficients.

Comments

The original reference is [a1]. Good modern accounts of Lefschetz duality can be found in [a2] and (from the point of view of sheaf cohomology) in [a3].

References

[a1]	S. Lefschetz, "Manifolds with a boundary and their transformations" Trans. Amer. Math. Soc. , 29 (1927) pp. 429–462 Zbl 53.0552.04
[a2]	C.R.F. Maunder, "Algebraic topology" , Cambridge Univ. Press, reprint (1980)
[a3]	B. Iversen, "Cohomology of sheaves" , Springer (1986)