Is the map $\Phi$ in the Mayer-Vietoris sequence for homology injective? Is $\Psi$ surjective? Consider the Mayer-Victoris for homology:
$$ \rightarrow H_n(A \cap B) \xrightarrow{\Phi} H_n(A) \oplus H_n(B) \xrightarrow{\Psi} H_n(X) \rightarrow$$
where $\Phi$ is the induced map of $\phi(x) = (x, -x)$ and $\Psi$ is the induced map of $\psi(x,y) = x+y$.  Clearly, $\phi$ is injective.  Is $\Phi$ also injective?  I have looked at this Mathematics StackExchange post, which tells me the answer for induced maps in general, but I am having trouble convincing myself of the existence or nonexistence of an appropriate right inverse for $\phi$.
Also, is $\psi$ supposed to be surjective?  I vaguely recall something from class to this effect...  If so, is $\Phi$ surjective?
 A: The maps $\Phi$ and $\Psi$ are typically neither injective nor surjective.   For a simple example, let $X=S^{n+1}\sqcup S^n\sqcup S^n$, let $A$ be the union of the northern hemisphere of $S^{n+1}$, the first copy of $S^n$, and the northern hemisphere of the second copy of $S^n$, and let $B$ be the the union of the southern hemisphere of $S^{n+1}$, the first copy of $S^n$, and the southern hemisphere of the second copy of $S^n$.
Then $A\cap B=S^n\sqcup S^n\sqcup S^{n-1}$, but $\Phi$ kills the generator of $H^n(A\cap B)$ coming from the first $S^n$ since it is nullhomotopic in both $A$ and $B$.  Also, $\Phi$ is not surjective since $H^n(A)\oplus H^n(B)$ has two copies of the second $S^n$.
The two copies of the second $S^n$ in $H^n(A)\oplus H^n(B)$ also make $\Psi$ not injective.  But $\Psi$ is not surjective either, since it does not hit the generator of $H^n(X)=H^n(S^{n+1}\sqcup S^n\sqcup S^n)$ coming from the second $S^n$.
On the chain level, the map $\psi:C_n(A)\oplus C_n(B)\to C_n(X)$ is typically not surjective (assuming you're talking about singular chains): there can be singular simplices in $X$ that are not contained in either $A$ or $B$.  However, it is "surjective up to homotopy", in that the inclusion of the image of $\psi$ into $C_n(X)$ is a chain homotopy equivalence.
