Math 6510 • Differentiable Manifolds

Let \(p:\widetilde{X}\to X\) be an \(n\)-sheeted covering map between two
connected, locally path-connected spaces. Let \(C_*(X)\) and
\(C_*(\widetilde{X})\) denote the singular chain complexes of \(X\) and
\(\widetilde{X}\) respectively.
\begin{enumerate}
\item[(a)] Give a definition of a chain morphism and prove that
\(\tau:C_*(X)\to C_*(\widetilde{X})\) given by
\[
  \tau(\sigma)=\sum_{i=1}^n \widehat{\sigma}_i
\]
is a chain morphism, where \(\widehat{\sigma}_1,\ldots,\widehat{\sigma}_n\) are
the \(n\) lifts of \(\sigma\) to \(\widetilde{X}\), and \(\sigma\) is a singular simplex
in \(X\).
\item[(b)] Prove that
\[
  p_*:H_i(\widetilde{X};\Q)\to H_i(X;\Q)
\]
is always surjective. Hint: Consider the composition
\(p_*\tau:C_*(X)\to C_*(X)\).
\item[(c)] Find an example where
\[
  p_*:H_i(\widetilde{X};\Z)\to H_i(X;\Z)
\]
is not surjective.
\end{enumerate}