I have a CW-complex $X$ with a closed subset $C$ and its complement $U$. Like in this similar question, I am working with the following long exact sequence of cohomology with compact supports:
$$ \cdots \to H^{i-1}_c (X) \to H^{i-1}_c (C) \to H^i_c(U) \to H^i_c (X) \to H^i_c (C) \to H^{i+1}_c (U) \to \cdots$$
I additionally know that two of the terms are trivial and can therefore form the following exact sequence:
$$ 0 \to H^{1}_c (C) \xrightarrow{i} H^2_c(U) \to H^2_c (X) \xrightarrow{r} H^2_c (C) \xrightarrow{t}\ 0$$
Does it follow that $H^{1}_c (C)=0$?
Here is the reason I think so. We have a short exact sequence: $$ 0 \to \ker{r} \to H^2_c (X) \xrightarrow{r} H^2_c (C) \xrightarrow{t} 0.$$
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 (C)- \{0\}) \\ & \cong H^2_c (X) - (H^2_c (C)- \{0\}) \\ & \cong H^2_c (U). \end{split} \end{equation}
I assume I am in error because the result is too strong, and I also assume the error is the last isomorphism. All wisdom is appreciated!