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Question

Math 6510 • Differentiable Manifolds

Fall 2001 • Question 3

Question

Math 6510 • Differentiable Manifolds Fall 2001 • Question 3

By \(a:S^n\to S^n\) denote the antipodal map. Then the space \(A^p(S^n)\) of smooth \(p\)-forms on \(S^n\) breaks up into the \(+1\) and the \(-1\) eigenspace of \(a^*\), i.e. \[ A^p(S^n)=A^p_+(S^n)\oplus A^p_-(S^n) \] where \[ A^p_\epsilon (S^n)=\{\omega \in A^p(S^n)\mid a^*(\omega )=\epsilon \omega \}. \] Show that if \(\omega \) is an exact \(p\)-form and belongs to the \(+1\)-eigenspace, then \(\omega =d\eta \) for some \((p-1)\)-form \(\eta \) in the \(+1\)-eigenspace. What can you conclude about DeRham cohomology \(H^i(\mathbb {R}P^n)\) from this?

By \(a:S^n\to S^n\) denote the antipodal map. Then the space \(A^p(S^n)\) of smooth
\(p\)-forms on \(S^n\) breaks up into the \(+1\) and the \(-1\) eigenspace of \(a^*\), i.e.
\[
  A^p(S^n)=A^p_+(S^n)\oplus A^p_-(S^n)
\]
where
\[
  A^p_\epsilon(S^n)=\{\omega\in A^p(S^n)\mid a^*(\omega)=\epsilon\omega\}.
\]
Show that if \(\omega\) is an exact \(p\)-form and belongs to the \(+1\)-eigenspace, then
\(\omega=d\eta\) for some \((p-1)\)-form \(\eta\) in the \(+1\)-eigenspace. What can you
conclude about DeRham cohomology \(H^i(\mathbb{R}P^n)\) from this?

Topic Tags

Cohomology and Cup Products Fixed Point Theorems and Borsuk-Ulam Real Projective Spaces de Rham Cohomology