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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2023 • Question 6

Question

Math 6510 • Differentiable Manifolds Fall 2023 • Question 6

Let \(\omega \) be a closed \(n\)-form on a smooth manifold \(M\) . Let \(N\) be a compact, orientable \(n\)-manifold without boundary and \(f:N\to M\) a smooth map with

\[ \int _N f^*\omega \ne 0. \]

Show that deRham cohomology group \(H^n_{dR}(M)\) is not trivial.

Raw LaTeX

Let \(\omega\) be a closed \(n\)-form on a smooth manifold \(M\) . Let \(N\) be a compact, orientable
\(n\)-manifold without boundary and \(f:N\to M\) a smooth map with

\[
\int_N f^*\omega\ne 0.
\]

Show that deRham cohomology group \(H^n_{dR}(M)\) is not trivial.

Topic Tags

Differential Forms and Exterior Calculus Orientability and Volume Forms Smooth Manifolds and Embedded Submanifolds de Rham Cohomology