Extension rule for Poincare duality groups ($P{D}^n\text{-by-}P{D}^q⟹P{D}^{n+q}$)

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.