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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2013 • Question 13

Question

Math 6510 • Differentiable Manifolds Spring 2013 • Question 13

  1. The complex projective space \(\mathbb {C}\mathrm {P}^n\) can be obtained from \(\mathbb {C}\mathrm {P}^{n-1}\) by attaching a single \(2n\)-cell. Describe the attaching map.
  2. What is \(H^k(\mathbb {C}\mathrm {P}^n;\mathbb {Z})\)? Give a proof by induction on \(n\) using (a). State any theorems you are using.
  3. What is the ring structure on \(H^*(\mathbb {C}\mathrm {P}^n;\mathbb {Z})\)? Give an inductive proof. You may use Poincaré duality.

Raw LaTeX

\begin{enumerate}
\item[(a)] The complex projective space \(\CP^n\) can be obtained from
\(\CP^{n-1}\) by attaching a single \(2n\)-cell. Describe the attaching map.
\item[(b)] What is \(H^k(\CP^n;\Z)\)? Give a proof by induction on \(n\) using (a).
State any theorems you are using.
\item[(c)] What is the ring structure on \(H^*(\CP^n;\Z)\)? Give an inductive
proof. You may use Poincar\'e duality.
\end{enumerate}

Topic Tags

CW Complexes and Cell Attachments Cohomology and Cup Products Poincare Duality Real Projective Spaces