I tried to understand the proof that Mayer-Vietoris short exact sequence is exact. In the internet there are a lot of proofs of that, but unfortunately in all of these proofs I couldn't understand one detail:
Mayer-Vietoris short exact sequence:
$0 \to \Delta_p(A \cap B) \to \Delta_p(A) \oplus \Delta_p(B) \to^{(j^A \pi_1-j^B \pi_2)} \Delta_p^U(A \cup B) \to 0$
I'm interested in the proof of surjectivity of the map $j^A \pi_1-j^B \pi_2$. Let $(\phi_A, \phi_B)$ be a partition of unity of subordinate to (A,B). Let $w \in \Delta_p^U(A \cup B)$. Then in every proof I saw we define $v= (\phi_A(w),-\phi_B(w)) $. Then $(j^A \pi_1-j^B \pi_2)(v)=j^A \phi_A(w)+j^B \phi_B(w)=(\phi_A+\phi_B)(w)=1w=w.$
The only problem with all this is that I don't understand why $\phi_A(w) \in \Delta_p(A)$. I looked at the domains of the functions:
$w:\Delta_p \to A\cap B$ and $\phi_A:A\cap B \to \Delta_p$ so I have $\phi_A w:\Delta_p \to \Delta_p$ and therefore $\phi_Aw$ is not an element of $\Delta_p(A)$
I know that my problem might be a very trivial one, but I'm confused because I can't see where I did a mistake with the domains of the functions. So I would appreciate your help.
First of all, $\phi_A$ should be a map $A\cup B\rightarrow [0,1]$. Second: I think you are trying to use the proof for the exactness of the Mayer-Vietoris sequence in de Rahm cohomology to prove the exactness of the Mayer-Vietoris sequence in simplicial or singular homology. Have you tried to take a look at Hatcher's book?