Question about Relative Cohomology

Question

I need help with the following question please:

Suppose that a space $X \subseteq Y $ retracts onto some subspace $A \subseteq X $.

When do I have $H^\ast ( Y,X) \cong H^\ast (Y,A)$?

Thanks.

Answer 1

This is false in general. Let $Y$ be a disk, $X$ be its boundary and $A$ be a point on the boundary. Then $H^2(Y, X) \cong \Bbb Z$ whereas $H^2(Y, A)$ is trivial.

Now if you mean $X$ deformation retracts onto $A$, then the statement is true. In this case, use the long exact sequence of the triple $(Y, X, A)$ and note that the inclusion $A \hookrightarrow X$ induces an isomorphism.