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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2009 • Question 8

Question

Math 6510 • Differentiable Manifolds Fall 2009 • Question 8

Let \(M=T^2\times S^2\), and suppose \(\omega \) is a 4-form on \(M\) such that \(\int _M\omega =7\).

  1. Define a smooth map \(f:M\to M\) that has degree 2.
  2. Define a 4-form \(\theta \) on \(M\) such that \(\int _M\theta =56\), using both \(f\) and \(\omega \) nontrivially.

Raw LaTeX

Let \(M=T^2\times S^2\), and suppose \(\omega\) is a 4-form on \(M\) such that
\(\int_M\omega=7\).
\begin{enumerate}
\item[(a)] Define a smooth map \(f:M\to M\) that has degree 2.
\item[(b)] Define a 4-form \(\theta\) on \(M\) such that \(\int_M\theta=56\), using both
\(f\) and \(\omega\) nontrivially.
\end{enumerate}

Topic Tags

Degree Theory Differential Forms and Exterior Calculus Homotopy and Degree on Manifolds