Math 6510 • Differentiable Manifolds

Two covering spaces \(p_0:X_0\to X\) and \(p_1:X_1\to X\) of a space \(X\) are
isomorphic if there is a homeomorphism \(h:X_0\to X_1\) such that \(p_0=p_1\circ h\). In
this question all spaces are path-connected, locally path-connected and semilocally
path-connected.
\begin{enumerate}
\item Prove the following piece of the classification theorem for covering spaces: Let
\(x\in X\), \(x_0\in X_0\) and \(x_1\in X_1\) be basepoints such that
\(p_0(x_0)=p_1(x_1)=x\). The covers are isomorphic if and only if
\((p_0)_*(\pi_1(X_0,x_0))\) and \((p_1)_*(\pi_1(X_1,x_1))\) are conjugate subgroups of
\(\pi_1(X,x)\).
\item Show that there are non-isomorphic covers of a genus two surface of every degree.
\end{enumerate}