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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Summer 2018 • Question 8

Question

Math 6510 • Differentiable Manifolds Summer 2018 • Question 8

Let \(n\ge 1\) be a positive integer.

  1. Give an elementary description of the universal cover \(X\to \mathbb {R}\mathrm {P}^n\) of \(\mathbb {R}\mathrm {P}^n\).
  2. Describe a CW structure on \(\mathbb {R}\mathrm {P}^n\).
  3. Compute the cohomology ring \(H^*(\mathbb {R}\mathrm {P}^n;\mathbb {Z}/2)\) and the map (from (a)) \(H^*(\mathbb {R}\mathrm {P}^n;\mathbb {Z}/2)\to H^*(X;\mathbb {Z}/2)\).

Raw LaTeX

Let \(n\ge1\) be a positive integer.
\begin{enumerate}
\item Give an elementary description of the universal cover \(X\to\RP^n\) of
\(\RP^n\).

\item Describe a CW structure on \(\RP^n\).

\item Compute the cohomology ring \(H^*(\RP^n;\Z/2)\) and the map (from (a))
\(H^*(\RP^n;\Z/2)\to H^*(X;\Z/2)\).
\end{enumerate}

Topic Tags

Cohomology and Cup Products Covering Spaces Homology with Coefficients Real Projective Spaces