Relationship between the fundamental classes of a group and a subgroup

Abstract Let 𝐺 be a group of type FP and cohomological dimension 𝑛. Let 𝐻 be a subgroup of 𝐺 (not necessarily of finite index), also of type FP and cohomological dimension 𝑛. Let 𝑅 be a commutative ring with identity 1 ≠ 0 1 0. Let D G = H n ⁢ (G, R ⁢ G) D₆=H^n (G, RG) and D H = H n ⁢ (H, R ⁢ H) D₇=H^n (H, RH). We define a map β: H n ⁢ (G, D G) → H n ⁢ (H, D H) H₍ (G, D₆) H₍ (H, D₇) and prove that 𝛽 sends the fundamental class of 𝐺 to the fundamental class of 𝐻. The definition of 𝛽 depends on first defining a map α H: D G → D H ₇ D₆ D₇, and the proof of the theorem on the fundamental classes depends on showing that α H ₇ is surjective. When both 𝐺 and 𝐻 are duality groups, we tie together the restriction map on cohomology, the duality isomorphisms, and a generalization of 𝛽.