Math 6510 • Differentiable Manifolds

Let \(p:\widetilde{X}\to X\) be a 2-sheeted covering map between cell complexes
such that the cell structure on \(\widetilde{X}\) is obtained by lifting the cell
structure on \(X\).
\begin{enumerate}
\item Construct a short exact sequence of cellular chain complexes with coefficients
in \(\mathbb{Z}/2\mathbb{Z}\)
\[
  0\to C_n(X;\mathbb{Z}/2\mathbb{Z})\to C_n(\widetilde{X};\mathbb{Z}/2\mathbb{Z})
  \to C_n(X;\mathbb{Z}/2\mathbb{Z})\to 0
\]
(i.e. give the maps explicitly and prove that the sequence is exact).
\item Write down the induced long exact sequence of homology groups (the transfer
sequence).
\item Prove that if \(\widetilde{X}\) is acyclic over \(\mathbb{Z}/2\mathbb{Z}\) (i.e.
\(H_i(\widetilde{X};\mathbb{Z}/2\mathbb{Z})=0\) for \(i>0\) and
\(H_0(\widetilde{X};\mathbb{Z}/2\mathbb{Z})\cong\mathbb{Z}/2\mathbb{Z}\)) then
\(H_i(X;\mathbb{Z}/2\mathbb{Z})\cong\mathbb{Z}/2\mathbb{Z}\) for all \(i\ge0\).
\end{enumerate}