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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2008 • Question 3

Question

Math 6510 • Differentiable Manifolds Fall 2008 • Question 3

Let \(M\) and \(N\) be compact orientable \(n\)-dimensional manifolds and \(f:M\to N\) a smooth map. Suppose that \[ \int _M f^*\Omega \ne 0 \] for some \(n\)-form \(\Omega \). Prove that \(f\) is surjective.

Raw LaTeX

Let \(M\) and \(N\) be compact orientable \(n\)-dimensional manifolds and
\(f:M\to N\) a smooth map. Suppose that
\[
  \int_M f^*\Omega \ne 0
\]
for some \(n\)-form \(\Omega\). Prove that \(f\) is surjective.

Topic Tags

Differential Forms and Exterior Calculus Homotopy and Degree on Manifolds Orientability and Volume Forms Smooth Manifolds and Embedded Submanifolds