My goal is to understand a particular short exact sequence derived from a Mayer-Vietoris Sequence. I will tell you what I have so far, and then I am stuck in defining the last group.
We begin with a contractible, infinite, one-ended 2-D simplicial complex X. Then we have a closed but infinite 2-D one-ended subset of X called V. Next we have $\overline{V^c}$, where $V \cap \overline{V^c}$ is the 1-D boundary of V.
This situation yields the following Mayer Vietoris Sequence:
$$ \cdots \to H^{1}_c (X) \to H^{1}_c (V) \oplus H^{1}_c (\overline{V^c}) \to H^1_c(V \cap \overline{V^c}) \xrightarrow{\delta} H^2_c (X) \to H^{2}_c (V) \oplus H^{2}_c (\overline{V^c}) \to H^2_c(V \cap \overline{V^c}) \to H^3_c (X) \to \cdots$$
The previous sequence is the limit over increasing compact sets $K \subset V$ and $L \subset \overline{V^c}$ of the following sequence:
$$ \cdots \to H^{1}_c (V \cup \overline{V^c} | K \cup L) \to H^{1}_c (V| K) \oplus H^{1}_c (\overline{V^c} | L) \to H^1_c(V \cap \overline{V^c} | K \cap L) \xrightarrow{\delta} H^2_c (V \cup \overline{V^c} | K \cup L) \to H^{2}_c (V|K) \oplus H^{2}_c (\overline{V^c}|L) \to H^2_c(V \cap \overline{V^c}|K \cap L) \to H^3_c (V \cup \overline{V^c} | K \cup L) \to \cdots$$
which is equivalent to this sequence:
$$ \cdots \to H^{1}_c (X | K \cup L) \to H^{1}_c (X| K) \oplus H^{1}_c (X | L) \to H^1_c(X | K \cap L) \xrightarrow{\delta} H^2_c (X | K \cup L) \to H^{2}_c (X|K) \oplus H^{2}_c (X|L) \to H^2_c(X |K \cap L) \to H^3_c (X | K \cup L) \to \cdots$$
The coboundary map $\delta$ arises from the following short exact sequence where $A=X-K$ and $B=X-L$.
$$0 \to C^*(X, A \cap B) \to C^*(X,A) \oplus C^*(X,B) \to C^*(X, A+B) \to 0$$
The first group is the set of cochains in X that vanish on chains completely outside of $K \cup L$. The second is the sum of two groups that consist of cochains that vanish on chains completely outside K and L, respectively. The last group of the sequence consists of cochains in X that vanish on chains that do not go through both K and L.
Now for completely unrelated reasons, it is also known in this particular situation that some of the terms in the original MV sequence are trivial, so we have the following:
$$ 0 \to H^{1}_c (V) \oplus 0 \xrightarrow{i} H^1_c(V \cap \overline{V^c}) \xrightarrow{\delta} H^2_c (X) \xrightarrow{r} H^{2}_c (V) \oplus 0 \to 0$$
The goal
The goal is to understand and define the first group in a short exact sequence that is derived from the previous exact sequence.
One such short exact sequence is the following:
$$ 0 \to \text{im}{\delta} \to H^2_c (X) \to H^{2}_c (V) \oplus 0 \to 0$$
So what needs to be done is to understand $\text{im}{\delta}$.
Attempt #1
My first attempt was to begin with the map $H^1_c(V \cap \overline{V^c}) \xrightarrow{\delta} H^2_c (X)$
Then we have the following: \begin{equation} \begin{split} \text{im}{\delta} & = \delta(H^1_c(V \cap \overline{V^c}) - \ker \delta)\\ & = \delta(H^1_c(V \cap \overline{V^c}) - \text{im}{i}) \\ & = \delta(H^1_c(V \cap \overline{V^c}) - i(H^{1}_c (V)-\ker i)) \\ & = \delta(H^1_c(V \cap \overline{V^c}) - i(H^{1}_c (V))) \\ & \cong \delta(H^1_c(V \cap \overline{V^c}) - H^{1}_c (V)) + {0} \\ \end{split} \end{equation}
Then we can note that $H^1_c(V \cap \overline{V^c})$ are cocycles in $V \cap \overline{V^c}$ that can only be nonzero on relative 1-cycles in $V \cap \overline{V^c}$ (since $V \cap \overline{V^c}$ is 1-dimensional.) And $H^{1}_c (V)$ are the cocycles in V that can only be nonzero on relative 1-cycles in V that are not filled by 2-simplices in V.
Then $H^1_c(V \cap \overline{V^c}) - H^{1}_c (V)$ are the cocycles in $V \cap \overline{V^c}$ that can only be nonzero on relative 1-cycles in $V \cap \overline{V^c}$ that have filling disks $D \subseteq V$.
That is,
$$\text{im}{\delta} = \delta(\{ \gamma \in H^1_c(V \cap \overline{V^c}) \text{ s.t. } \gamma(\sigma)=0 \text{ for every } \sigma, \text{ a relative loop in } \overline{V^c} \text{ that has no filling disk } D \subset V \}).$$
Then $\text{im}\delta$ are the 2-cocycles that contain the 1-cocycles $\{ \gamma \in H^1_c(V \cap \overline{V^c}) \text{ s.t. } \gamma(\sigma)=0 \text{ whenever } \sigma \text{ is a relative loop in } \overline{V^c} \text{ that has no filling disk } D \subset V \}$.
That is as far as I got with this method, and it's probably the closest to a definition of $\text{im}{\delta}$ I have reached (besides Attempt #4), but is there a way to make this more explicit?
Attempt #2
My next attempt was to consider $\text{im}{\delta} = \ker{r}$. Then \begin{equation} \begin{split} \ker{r} & = H^2_c (X) - r^{-1}(\text{im} r - \{0\}) \\ & = H^2_c (X) - r^{-1}(\text{ker} t - \{0\}) \\ & = H^2_c (X) - r^{-1}(H^2_c (V)- \{0\}) \\ \end{split} \end{equation}
Next I wanted to replace $r^{-1}(H^2_c (V)- \{0\})$ with $H^2_c (V)- \{0\}$, but as FShrike pointed out, $H^2_c (V)$ is not a subgroup of $H^2_c (X)$. So this attempt ended there.
Attempt #3
My third attempt was to go back to the cochain complex above to define $\text{im}{\delta}$. Here we define $\gamma \in C^1(X, A+B)$ as $\gamma = \gamma_A - \gamma_B$. For $\gamma_A \in C^1(X, A)$ and $\gamma_B \in C^1(X, B)$. Then $\delta \gamma$ is represented by $\delta \gamma_A = \delta \gamma_B \in C^2(X, A \cap B)$.
Attempt #4
(This seems to be the best method so far.)
An alternative way to form a short exact sequence is using the cokernel. That is, $$ 0 \to \text{coker}{i} \to H^2_c (X) \to H^{2}_c (V) \oplus 0 \to 0$$
This method would then require defining $\text{coker}{i} = H_c^1(V \cap \overline{V^c})/\text{im}i.$ By the analysis in Attempt 1, this would mean that $\text{coker}i$ are the 1-cocycles $\{ \gamma \in H^1_c(V \cap \overline{V^c}) \text{ s.t. } \gamma(\sigma)=0 \text{ whenever } \sigma \text{ is a relative loop in } \overline{V^c} \text{ that has no filling disk } D \subset V \}$.
This method makes me suspicious because short exact sequences using kernels are more common than those from cokernels.
Final Note
I've only written a portion of my work (and unfortunately I'm typing on a smartphone), but I'm hoping there will be some mathematician or teacher here that can set me on the right path to understand $\text{im}{\delta}$ or (better yet) $\text{coker}{i}$ so that I have a short exact sequence for which I can define all three terms. All insight would be so appreciated!