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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2009 • Question 6

Question

Math 6510 • Differentiable Manifolds Spring 2009 • Question 6

Let \(\Lambda ^k(M)\) denote the space of smooth \(k\)-forms on a manifold \(M\). Suppose that \(\omega \in \Lambda ^p(M)\) and \(\eta \in \Lambda ^q(M)\) are closed forms. Show that the de Rham cohomology class of \(\omega \wedge \eta \) depends only on the de Rham cohomology classes of \(\omega \) and \(\eta \).

Raw LaTeX

Let \(\Lambda^k(M)\) denote the space of smooth \(k\)-forms on a manifold \(M\).
Suppose that \(\omega\in\Lambda^p(M)\) and \(\eta\in\Lambda^q(M)\) are closed forms.
Show that the de Rham cohomology class of \(\omega\wedge\eta\) depends only on the
de Rham cohomology classes of \(\omega\) and \(\eta\).

Topic Tags

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