Math 6520 • Algebraic Topology

Consider the group \(G\) given by the presentation

\[
G=\langle a,b\mid a^p b^q=1\rangle
\]


where \(p\), \(q\) are given integers.
\begin{enumerate}

\item[(i)]
Construct a (connected) CW-complex \(X\) with a basepoint \(x\) such
that \(\pi_1(X,x)\cong G\).

\item[(ii)]
Compute \(H_1(X)\) and give conditions on \(p\), \(q\) that are equivalent to
\(H_1(X)\) being torsion-free.

\end{enumerate}