Math 6510 • Differentiable Manifolds

Let \(m,n\) be positive integers and view \(S^1\) as the unit circle in the
complex plane. Let \(X\) be the space obtained from the cylinder \(S^1\times[0,1]\)
by identifying points \((z,0)\) and \((w,0)\) if \(z^n=w^n\), and identifying points
\((z,1)\) and \((w,1)\) if \(z^m=w^m\).
\begin{enumerate}
\item Use the Mayer-Vietoris sequence to compute \(H_i(X)\) for all \(i\).

\item Use the Seifert-van Kampen theorem to find a presentation for
\(\pi_1(X)\).
\end{enumerate}