# rational homology manifolds

A compact, complex algebraic variety $X$ is said to be a "rational homology manifold" if, for any point $x \in X$, $H^i_\{x\}(X, \mathbb{Q})=\mathbb{Q}$ if $i=2\dim X$ and zero otherwise. How does one prove Poincaré duality for this class of manifold? Is there a good reference for them?

• Surely you don't want to assume "manifold" (much less "complex") to begin with. – Allen Knutson Jun 22 '13 at 14:17
• I wanted to say compact algebraic variety over the complex numbers. – damien Jun 22 '13 at 14:32
• First of all, isn't this false? It's true that an integral homology manifold is a rational homology manifold, but not all such manifolds are oriented. Assuming existence of an orientation (and this is certainly far from the original reference), Hatcher's "Algebraic Topology," starting in the section "Orientations and Homology" in section 3.3, and culminating in the statement of Poincare duality in Theorem 3.30 gives what you're after (let $R = \mathbb{Q}$). – Craig Westerland Sep 24 '13 at 1:45