Math 6520 • Algebraic Topology

\begin{enumerate}

\item
Construct a single chain complex \(C\) such that \(C\) has the following
three properties:
\begin{itemize}
\item \(H_i(C)\) is non-zero if and only \(i=1\).
\item There is a coefficient group \(G_0\) such that \(\tilde{H}^i(C;G_0)=0\) for
all \(i\).
\item There is a coefficient group \(G_1\) such that \(\tilde{H}^i(C;G_1)\) is non-
zero if and only if \(i=1,2\).
\end{itemize}


\item
Find a topological space \(X\) such that \(H_i(C)=\tilde{H}_i(X)\) for all \(i\).

\end{enumerate}