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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2019 • Question 8

Question

Math 6510 • Differentiable Manifolds Spring 2019 • Question 8

  1. Explain how the set of lines in \(\mathbb {C}^{n+1}\) is topologized to yield the space \(\mathbb {C}\mathrm {P}^n\). Describe how to equip \(\mathbb {C}\mathrm {P}^n\) with a CW structure with \(m\)-skeleton isomorphic to \(\mathbb {C}\mathrm {P}^m\) for all \(m\le n\).
  2. Compute the singular homology groups \(H_*(\mathbb {C}\mathrm {P}^n;\mathbb {Z})\). Be sure you indicate which properties of homology you are using.

Raw LaTeX

\begin{enumerate}
\item[(a)] Explain how the set of lines in \(\C^{n+1}\) is topologized to yield
the space \(\CP^n\). Describe how to equip \(\CP^n\) with a CW structure with
\(m\)-skeleton isomorphic to \(\CP^m\) for all \(m\le n\).
\item[(b)] Compute the singular homology groups \(H_*(\CP^n;\Z)\). Be sure you
indicate which properties of homology you are using.
\end{enumerate}

Topic Tags

Cellular Homology Homology with Coefficients Relative Homology and Long Exact Sequences