Math 6510 • Differentiable Manifolds

Fix integers \(n\ge1\) and \(m\ge1\).

\begin{enumerate}
\item[(a)] Construct a connected space \(X\) such that
\[
  \widetilde{H}_r(X;\Z)=
  \begin{cases}
    0 & \text{if } r\ne n,\\
    \Z/m & \text{if } r=n.
  \end{cases}
\]
(Here \(\widetilde{H}\) denotes reduced homology.)
\item[(b)] For any connected space \(X\) satisfying the conclusion of part (a),
compute the groups \(H_r(X;\Z/2)\) for all \(r\ge0\).
\end{enumerate}