# Show $H_n(X_n,\,X_{n-1}) \cong H_n(X_n/X_{n-1}, \ast)$

Assume $X_{-1}:=\emptyset, X_0 \subset X_1 \subset \dots \subset X\;$ is a CW-complex with pushouts


where '$inc$' denotes the set-inclusion in all cases.

How do I show that $$H_n(X_n,\,X_{n-1}) \cong H_n(X_n/X_{n-1}, \ast)$$ is an isomorphism of groups? Note that $H_n$ denotes the $n$-th functor of singular homology. Also, the $\ast$ is shorthand for $\{X_{n-1}\}$, i.e. the class of $X_{n-1}$ under the canonical projection.

I tried using the LESes for the respective pairs, connecting them through the projection-induced morphisms. But I don't think that was enough...

I am studying for an exam and would appreciate if someone could help me with this, complete answers are appreciated. This isomorphy is part of a proof in the lecture notes, but the professor didn't bother to make a remark about it.

• Doesn't this just follow from the fact that $(X_n , X_{n-1})$ is a good pair? – Kenny Wong Apr 19 '17 at 19:47
• I see it now, thank you! Do you want to post an answer (if not, I will do that)? – polynomial_donut Apr 19 '17 at 19:51
• Feel free to post an answer yourself. :) – Kenny Wong Apr 19 '17 at 19:51
• Okay, thank you! Great help :) – polynomial_donut Apr 19 '17 at 19:52

