Math 6510 • Differentiable Manifolds

Let \(X\) be the space obtained from the circle \(S^1\) by attaching two 2-cells,
one with degree 3 attaching map and the other with degree 5 attaching map.
\begin{enumerate}
\item[(a)] Using van Kampen's theorem show that \(X\) is simply connected.
\item[(b)] Show that \(X\) is homotopy equivalent to the 2-sphere. Hint: Use
repeateadly the fact that the spaces \(Y\cup_f e^2\) and \(Y\cup_g e^2\) obtained from
\(Y\) by attaching a single 2-cell with homotopic attaching maps \(f,g\) are homotopy
equivalent.
\end{enumerate}