View source for Lefschetz duality

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''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 [[#References|[a1]]]. Good modern accounts of Lefschetz duality can be found in [[#References|[a2]]] and (from the point of view of sheaf cohomology) in [[#References|[a3]]].

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Lefschetz,   "Manifolds with a boundary and their transformations"  ''Trans. Amer. Math. Soc.'' , '''29'''  (1927)  pp. 429–462 {{ZBL|53.0552.04}}</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  C.R.F. Maunder,   "Algebraic topology" , Cambridge Univ. Press, reprint  (1980)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  B. Iversen,   "Cohomology of sheaves" , Springer  (1986)</TD></TR></table>