Math 6510 • Differentiable Manifolds

Let \(X\) be the compact surface
\[
  X=\left\{
    z\in\C \mid |z|\le1,\,
    \left|z-\frac12\right|\ge\frac14,\,
    \left|z+\frac12\right|\ge\frac14
  \right\}
\]
obtained from the disk by deleting two disjoint open disks. There are two
essentially different ways to identify the three boundary components by
homeomorphisms: fix orientations on two boundary components as in the Figure below
and use two possible orientations on the third. Show that the two quotient spaces
have non-isomorphic fundamental groups. Hint: Abelianize.
\begin{center}
  \includegraphics[
    alt={Disk with two interior boundary circles; the two interior boundary components have indicated arrows with matching counterclockwise orientations.},
    width=0.38\textwidth
  ]{GeomTopF2014-diagram-9.png}
\end{center}