TITLE: Compute the homology groups using Mayer-Vietoris sequence
QUESTION [5 upvotes]: I need to compute the homology groups under group of integers $ H_k(D; \mathbb Z) $, of the simplicial complex being a triangulation of the following figure:

I divided it two parts $D = L_1 \cup L_2$ like this:

Then both parts $L_i$ are homeomorphic to the standard 2-dimentional simplicial complex and hence we know thier homology groups. 
$$ H_k(\Delta^2, \mathbb Z) = \begin{cases}
\mathbb Z, & k = 0 \\
0, & k > 0
\end{cases} $$
The intersection $L_1 \cap L_2 = I_1 \cup I_2 \cup I_3$ of the parts is merging of 3 disjoint segments and hence we know their homology groups too.
$$ H_k(I_1 \cup I_2 \cup I_3, \mathbb Z) = \begin{cases}
\mathbb Z \oplus \mathbb Z \oplus \mathbb Z, & k = 0 \\
0, & k > 0
\end{cases} $$
And we can construct the Mayer-Vietoris sequence:
$$ 0 \to H_1(L_1 \cap L_2; \mathbb Z) \to H_1(L_1; \mathbb Z) \oplus H_1(L_2; \mathbb Z) \to H_1(L_1 \cup L_2; \mathbb Z) \to H_0(L_1 \cap L_2; \mathbb Z) \to H_0(L_1; \mathbb Z) \oplus H_0(L_2; \mathbb Z) \to H_0(L_1 \cup L_2; \mathbb Z) \to 0 $$
And substituting computed groups:
$$ 0 \to 0 \to 0 \to H_1(D; \mathbb Z) \to \mathbb Z \oplus \mathbb Z \oplus \mathbb Z \to \mathbb Z \oplus \mathbb Z \to H_0(D; \mathbb Z) \to 0 $$
How can I compute from this groups $ H_0(D, \mathbb Z) $ and $ H_1(D, \mathbb Z) $? Thanks for the help!

REPLY [2 votes]: From this information alone, you cannot compute the groups, because they could be 
$$
\mathbb Z \oplus \mathbb Z \oplus \mathbb Z
$$
and 
$$
\mathbb Z \oplus \mathbb Z
$$
with the two obvious maps being isomorphisms, or the first could be $\mathbb Z$ and the second be $0$. There are surely other possibilities. 
To work out the groups, you'd need to know the actual maps on the various groups. 
It's considerably easier to divide the shape with a line running NW-SE rather than NE/SW. Then you get two circles, whose homology groups you know, and the intersection is a line segment, which is contractible. And the map from $H_0$ 
of the intersection to $H_0$ of either part is an isomorphism (generated by the map taking a  0-simplex to a point in the intersection).