Math 6510 • Differentiable Manifolds

Let \(p,q\ge2\) be integers and denote by \(X_{p,q}\) the join of discrete sets
\(\{v_1,\ldots,v_p\}\) and \(\{w_1,\ldots,w_q\}\). Thus \(X_{p,q}\) is a graph with
\(p+q\) vertices and \(pq\) edges.
\begin{enumerate}
\item[(a)] What is the fundamental group of \(X_{p,q}\)?
\item[(b)] Denote by \(Y_{p,q}\) the 2-complex obtained from \(X_{p,q}\) by attaching
2-cells with attaching maps \(S^1\to X_{p,q}\) homeomorphisms to all possible
cycles of length 4. (Thus there are \(\binom{p}{2}\binom{q}{2}\) 2-cells.) Show
that \(Y_{p,q}\) is simply connected.
\end{enumerate}