Math 6510 • Differentiable Manifolds

Let \(X_n\) be the space obtained by attaching a disk \(D\) to the torus
\(T=S^1\times S^1\) where the attaching map is a degree \(n\) map from \(\partial D\) to
the circle \(S^1\times\{p\}\) in \(T\).
\begin{enumerate}
\item[(a)] Calculate \(\pi_1(X_n)\).
\item[(b)] Calculate the homology and cohomology of \(X_n\) with \(\Z\)-coefficients
and with \(\Z_m\)-coefficients where \(\Z_m\) is the cyclic group of order \(m\).
\end{enumerate}