Mayer-Vietoris sequence in simplical homology for a $\Delta$-complex Hatcher's Exercise 2.2.37 asks as follows:
Exercise 2.2.37. Give an elementary derivation for the Mayer-Vietoris sequence in simplical homology for a $\Delta$-complex $X$ decomposed as the union of subcomplexes $A$ and $B$.
I don't understand what this question exactly means. It seems like we can derive the M-V sequences in the special case for $\Delta$-complexes more easily than the general case, but I have no idea. How can I derive the M-V seq. for $\Delta$-complexes in an elementary way?
 A: Perhaps it might help to consider that this is not about a "special case" of the Mayer-Vietoris of singular homology. It is instead the Mayer-Vietoris sequence of simplicial homology.
You are being asked to figure out what "the Mayer-Vietoris sequence in simplicial homology" actually means, so you need to define the terms and the arrows of that sequence, and combine those appropriately to form a "Mayer-Vietoris sequence". 
Having done that, your job is then to prove that the sequence you have written down is exact.
Regarding the meaning, the idea is simple: for each subcomplex $Y \subset X$, restrict the $\Delta$-complex structure on $X$ to obtain a $\Delta$-complex structure of $Y$, use that to define the simplicial chain complex $\Delta_n(Y)$, and use that to define its simplicial homology groups $H_n(X)$; also, use the inclusion $Y \subset X$ to define the chain maps $\Delta_n(Y) \to \Delta_n(X)$, and use that to define the induced homology homomorphisms $H_n(Y) \mapsto H_n(X)$. 
Now apply this to three subcomplexes: $Y=A$; $Y=B$; and $Y=A \cap B$. Once that's done, you've pretty much defined all the terms and all the arrows of the following sequence, subject to throwing in appropriate $\pm$ signs for the maps defining the arrows:
$$\cdots \mapsto H_n(A \cap B) \mapsto H_n(A) \oplus H_n(B) \mapsto H_n(X) \mapsto H_{n-1}(A \cap B) \mapsto\cdots
$$
and now prove that this sequence is exact.
