Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2019 • Question 3

Question

Math 6510 • Differentiable Manifolds Fall 2019 • Question 3

Let \(M\) and \(N\) be smooth manifolds. Without using the de Rham Theorem, show that if the de Rham cohomology of \(M\) doesn’t vanish \(H^k_{\mathrm {dR}}(M)\ne \{0\}\) then \(H^k_{\mathrm {dR}}(M\times N)\ne \{0\}\).

Raw LaTeX

Let \(M\) and \(N\) be smooth manifolds. Without using the de Rham Theorem, show
that if the de Rham cohomology of \(M\) doesn't vanish \(H^k_{\mathrm{dR}}(M)\ne\{0\}\)
then \(H^k_{\mathrm{dR}}(M\times N)\ne\{0\}\).

Topic Tags

Cohomology and Cup Products Homology with Coefficients Smooth Manifolds and Embedded Submanifolds de Rham Cohomology