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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2018 • Question 6

Question

Math 6510 • Differentiable Manifolds Spring 2018 • Question 6

Let \(M\) and \(N\) be smooth manifolds, and let \(f:M\to N\) be smooth. Prove that if \(\theta \) and \(\psi \) are closed differentiable \(n\)-forms on \(N\) (so that \([\theta ]\) and \([\psi ]\) are classes in de Rham cohomology) then \([\theta ]=[\psi ]\) implies \([f^*\theta ]=[f^*\psi ]\).

Raw LaTeX

Let \(M\) and \(N\) be smooth manifolds, and let \(f:M\to N\) be smooth. Prove
that if \(\theta\) and \(\psi\) are closed differentiable \(n\)-forms on \(N\) (so that
\([\theta]\) and \([\psi]\) are classes in de Rham cohomology) then \([\theta]=[\psi]\)
implies \([f^*\theta]=[f^*\psi]\).

Topic Tags

Cohomology and Cup Products Differential Forms and Exterior Calculus Smooth Manifolds and Embedded Submanifolds de Rham Cohomology