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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Winter 2006 • Question 7

Question

Math 6510 • Differentiable Manifolds Winter 2006 • Question 7

  1. Let \(a:S^n\to S^n\) be the antipodal map. Suppose that \(\omega \) is a smooth form on \(S^n\) such that \(a^*\omega =\omega \). Prove that if \(\omega \) is exact, then there is a smooth form \(\eta \) with \(\omega =d\eta \) and \(a^*\eta =\eta \).
  2. Use (a) and the fact that \(H^k_{\mathrm {DR}}(S^n)=0\) for \(0<k<n\) to deduce that \(H^k_{\mathrm {DR}}(\mathbb {R}P^n)=0\) for \(0<k<n\).

Raw LaTeX

\begin{enumerate}
\item Let \(a:S^n\to S^n\) be the antipodal map. Suppose that \(\omega\) is a smooth form
on \(S^n\) such that \(a^*\omega=\omega\). Prove that if \(\omega\) is exact, then there is a
smooth form \(\eta\) with \(\omega=d\eta\) and \(a^*\eta=\eta\).
\item Use (a) and the fact that \(H^k_{\mathrm{DR}}(S^n)=0\) for \(0<k<n\) to deduce that
\(H^k_{\mathrm{DR}}(\mathbb{R}P^n)=0\) for \(0<k<n\).
\end{enumerate}

Topic Tags

Cohomology and Cup Products Differential Forms and Exterior Calculus Fixed Point Theorems and Borsuk-Ulam Real Projective Spaces