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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2021 • Question 5

Question

Math 6510 • Differentiable Manifolds Fall 2021 • Question 5

Let \(M\) be a closed, compact manifold and show that if \(\omega \) is a nowhere zero 1-form on \(M\) then the de Rham cohomology group \(H^1_{dR}(M)\) is non-trivial. Use this to show that if there is a submersion from \(M\) to the circle then \(H^1_{dR}(M)\) is non-trivial.

Raw LaTeX

Let \(M\) be a closed, compact manifold and show that if \(\omega\) is a nowhere
zero 1-form on \(M\) then the de Rham cohomology group \(H^1_{dR}(M)\) is
non-trivial. Use this to show that if there is a submersion from \(M\) to
the circle then \(H^1_{dR}(M)\) is non-trivial.

Topic Tags

Differential Forms and Exterior Calculus Homotopy and Degree on Manifolds Immersions, Submersions, and Embeddings de Rham Cohomology