Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2012 • Question 15

Question

Math 6510 • Differentiable Manifolds Fall 2012 • Question 15

  1. Describe the simplest cell structures on \(\mathbb {C}\mathrm {P}^2\) (including attaching maps).
  2. Prove that \(H_i(\mathbb {C}\mathrm {P}^2;G)\) is isomorphic to \(H_i(S^2\vee S^4;G)\) for every abelian group \(G\) and all \(i\), and the same is true for cohomology.
  3. Prove that \(\mathbb {C}\mathrm {P}^2\) is not homotopy equivalent to \(S^2\vee S^4\) by considering ring structures in cohomology.

Raw LaTeX

\begin{enumerate}
\item[(a)] Describe the simplest cell structures on \(\CP^2\) (including attaching
maps).
\item[(b)] Prove that \(H_i(\CP^2;G)\) is isomorphic to \(H_i(S^2\vee S^4;G)\) for
every abelian group \(G\) and all \(i\), and the same is true for cohomology.
\item[(c)] Prove that \(\CP^2\) is not homotopy equivalent to \(S^2\vee S^4\) by
considering ring structures in cohomology.
\end{enumerate}

Topic Tags

CW Complexes and Cell Attachments Cohomology and Cup Products Homotopy Equivalence, Retracts, and Contractibility Homotopy and Degree on Manifolds