# at.algebraic topology – Invariants in relative cohomology and cohomology of compact support of the quotient

Leave $$cal H$$ be the Poincare upper half-plane and $$overline { cal H}$$ the Union of $$cal H$$ with the set of cusps $$bf P ^ 1 ( bf Q)$$, provided with its usual topology. Leave $$Gamma$$ a subgroup of congruence that acts freely on $$cal H$$, $$V$$ an Abelian group with $$Gamma$$-accion, and $$tilde V$$ The local system associated in the quotient. $$cal H / Gamma$$.

Ash and Stevens claim that there is a natural isomorphism
$$H ^ 1 ( overline { cal H}, { bf P ^ 1} ({ bf Q}), V) ^ Gamma = H ^ 1_c ({ cal H} / Gamma, tilde V).$$
This is in his paper in Duke, vol. 53, no 3, 1986, "MODULAR FORMS IN CHARACTERISTICS $$ell$$ AND SPECIAL VALUES OF YOUR L-FUNCTIONS ", page 862. They do not give any justification or proof.

Can anyone explain this isomorphism or point out the general results in the algebraic topology with which it can be demonstrated?