Isomorphism of homology chain groups I am having trouble tackling the following question.
Let $X$ be a simplicial complex. Suppose $X = B∪C$ for subcomplexes $B$ and $C$, and
let $A = B ∩ C$. Show that the inclusion of $A$ in $B$ induces an isomorphism $H_*A → H_∗B$
if and only if the inclusion of $C$ in $X$ induces an isomorphism $H_∗C → H_∗X$.
Suppose we assume there is an isomorphism $H_∗C → H_∗X$. My idea was to use Meyer-Vietoris, e.g
$$...\to H_0(A)\to H_0(B)\oplus H_0(C)\to H_0(X)\to0$$
and the fact that $H_0(C)\cong H_0(X)$ to show $H_0(A)\cong H_0(B)$ (and similarly for the other cases). However I am add loggerheads as how to do this. Any help or suggestions for better ways to solve this problem?
Edit:
Using the hint given below, we can get a short exact sequence of free abelian groups $$0\to H_0(A)\to H_0(B)\oplus H_0(C)\to H_0(X)\to0$$ from whence we can derive that $$ H_0(B)\oplus H_0(C)\cong  H_0(A)\oplus H_0(X)$$
Or equivalently $$ H_0(B)\oplus H_0(X)\cong  H_0(A)\oplus H_0(X)$$
from which it seems reasonable to derive $$ H_0(B)\cong  H_0(A)$$ However in general cancellation of groups fails, so what other information do we need?
 A: This answer is an elaboration of my comment.
For simplicity, let me abbreviate $H_*(A)$ to $A_*$ and similarly for the other homology groups. The splitting lemma yields, for a given section $s:X_*\to B_*\oplus C_*$, an isomorphism $A_*\oplus X_*\to B_*\oplus C_*$ given by $a\oplus x \mapsto \iota(a)+s(x)$. (where $\iota:A_*\to B_*\oplus C_*$ is a part of the short exact sequence)
In our case, we take the section to be the composition $X_*\cong C_*\hookrightarrow B_*\oplus C_*$, so after the identification $X_*=C_*$ the isomorphism becomes $A_*\oplus C_*\to B_*\oplus C_*, a\oplus c \mapsto \iota(a)+c$. To make it look more obvious, let me write it in matrix form:
$\begin{pmatrix}a\\c\end{pmatrix}\mapsto\begin{pmatrix}\iota_B & \iota_C\\0 & 1\end{pmatrix}\begin{pmatrix}a\\c\end{pmatrix}$
(where $\iota_B$ is the projection of $\iota$ onto $B_*$, etc)
Now it is clear that, as this upper triangular matrix represent an isomorphism, its diagonals must be an isomorphism. Especially, $\iota_B:A_*\to B_*\oplus C_*\to B_*$ is an isomorphism as desired.
