= Extension rule for Poincare duality groups
{title2=$PD^n\text{-by-}PD^q\Longrightarrow PD^{n+q}$}
For an extension with kernel a dimension-$n$ <Poincare duality group> and quotient a dimension-$q$ <Poincare duality group>, apply the <Lyndon–Hochschild–Serre spectral sequence> to the ambient group ring. Kernel cohomology is concentrated in degree $n$ and is a regular quotient module with a rank-one orientation twist. Quotient cohomology is then concentrated in degree $q$ with underlying group $\mathbb Z$. The extension resolution supplies finiteness and gives cohomological dimension at most $n+q$, proving duality in that dimension. Orientation actions need not be trivial.
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