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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2005 • Question 13

Question

Math 6510 • Differentiable Manifolds Fall 2005 • Question 13

  1. Describe the (simplest) cell structure of \(\mathbb {R}P^n\) (give the attaching maps of all the cells – you don’t have to give a proof here).
  2. Write down the cellular chain complex associated with this cell decomposition.
  3. Use this chain complex to compute the homology and cohomology groups of \(\mathbb {R}P^n\) with coefficients in \(\mathbb {Z}\) and \(\mathbb {Z}_2\).

Raw LaTeX

\begin{enumerate}
\item Describe the (simplest) cell structure of \(\mathbb{R}P^n\) (give the attaching
maps of all the cells -- you don't have to give a proof here).
\item Write down the cellular chain complex associated with this cell decomposition.
\item Use this chain complex to compute the homology and cohomology groups of
\(\mathbb{R}P^n\) with coefficients in \(\mathbb{Z}\) and \(\mathbb{Z}_2\).
\end{enumerate}

Topic Tags

CW Complexes and Cell Attachments Chain Complexes and Chain Homotopy Cohomology and Cup Products Homology with Coefficients