Math 6510 • Differentiable Manifolds

Below are pictured two graphs \(X,Y\) with basepoints \(x,y\) respectively and
a covering map \(f:X\to Y\) that takes edges to edges preserving labels and
orientation.

\newpage

\begin{center}
  \includegraphics[
    alt={A labeled covering graph diagram. The upper graph X has basepoint x and directed edges labeled a and b; the lower graph Y has basepoint y with two directed loops labeled a and b; a downward arrow labeled f maps x to y.},
    width=0.47\textwidth
  ]{GeomTopS2012-diagram-7.png}
\end{center}

Identify \(\pi_1(Y,y)\) with the free group with basis \(\{a,b\}\) as usual.
Describe the image of \(f_\#:\pi(X,x)\to\pi_1(Y,y)\).