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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2020 • Question 6

Question

Math 6510 • Differentiable Manifolds Spring 2020 • Question 6

Let \(M\) be a smooth compact manifold without boundary. Prove that if \(M\) has a smooth nowhere vanishing closed one form, then \(H^1_{\operatorname {dR}}(M)\ne \{0\}\). Deduce that if there is a smooth submersion of \(M\) onto \(S^1\), then \(H^1_{\operatorname {dR}}(M)\ne \{0\}\).

Raw LaTeX

Let \(M\) be a smooth compact manifold without boundary. Prove that if \(M\)
has a smooth nowhere vanishing closed one form, then \(H^1_{\dR}(M)\ne\{0\}\).
Deduce that if there is a smooth submersion of \(M\) onto \(S^1\), then
\(H^1_{\dR}(M)\ne\{0\}\).

Topic Tags

Cohomology and Cup Products Differential Forms and Exterior Calculus Orientability and Volume Forms Smooth Manifolds and Embedded Submanifolds