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Question

Math 6510 • Differentiable Manifolds

Spring 2008 • Question 13

Question

Math 6510 • Differentiable Manifolds Spring 2008 • Question 13

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 \(X\).

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).
Write down the induced long exact sequence of homology groups (the transfer sequence).
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 \(i\ge 0\).

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[(a)] Construct a short exact sequence of cellular chain complexes with
coefficients in \(\Z/2\Z\)
\[
  0\to C_n(X;\Z/2\Z)\to C_n(\widetilde{X},\Z/2\Z)
  \to C_n(X,\Z/2\Z)\to0
\]
(i.e. give the maps explicitly and prove that the sequence is exact).
\item[(b)] Write down the induced long exact sequence of homology groups (the
transfer sequence).
\item[(c)] Prove that if \(\widetilde{X}\) is acyclic over \(\Z/2\Z\) (i.e.
\(H_i(\widetilde{X};\Z/2\Z)=0\) for \(i>0\) and
\(H_0(\widetilde{X};\Z/2\Z)\cong\Z/2\Z\)) then
\(H_i(X;\Z/2\Z)\cong\Z/2\Z\) for all \(i\ge0\).
\end{enumerate}

Topic Tags

CW Complexes and Cell Attachments Cellular Homology Chain Complexes and Chain Homotopy Relative Homology and Long Exact Sequences