Relationship between the fundamental classes of a group and a subgroup

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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>1</m:mn> <m:mo>≠</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> 1≠ 0 . Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>D</m:mi> <m:mi>G</m:mi> </m:msub> <m:mo>=</m:mo> <m:mrow> <m:msup> <m:mi>H</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo>,</m:mo> <m:mrow> <m:mi>R</m:mi> <m:mo>⁢</m:mo> <m:mi>G</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> DG=Hⁿ(G,RG) and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>D</m:mi> <m:mi>H</m:mi> </m:msub> <m:mo>=</m:mo> <m:mrow> <m:msup> <m:mi>H</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>H</m:mi> <m:mo>,</m:mo> <m:mrow> <m:mi>R</m:mi> <m:mo>⁢</m:mo> <m:mi>H</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> DH=Hⁿ(H,RH) . We define a map <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>β</m:mi> <m:mo lspace="0.278em" rspace="0.278em">:</m:mo> <m:mrow> <m:mrow> <m:msub> <m:mi>H</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo>,</m:mo> <m:msub> <m:mi>D</m:mi> <m:mi>G</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">→</m:mo> <m:mrow> <m:msub> <m:mi>H</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>H</m:mi> <m:mo>,</m:mo> <m:msub> <m:mi>D</m:mi> <m:mi>H</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:math> β Hₙ(G,DG)→ Hₙ(H,DH) 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>α</m:mi> <m:mi>H</m:mi> </m:msub> <m:mo lspace="0.278em" rspace="0.278em">:</m:mo> <m:mrow> <m:msub> <m:mi>D</m:mi> <m:mi>G</m:mi> </m:msub> <m:mo stretchy="false">→</m:mo> <m:msub> <m:mi>D</m:mi> <m:mi>H</m:mi> </m:msub> </m:mrow> </m:mrow> </m:math> αH DG→ DH , and the proof of the theorem on the fundamental classes depends on showing that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>α</m:mi> <m:mi>H</m:mi> </m:msub> </m:math> αH 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.