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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2017 • Question 6

Question

Math 6510 • Differentiable Manifolds Spring 2017 • Question 6

Let \(M\) and \(N\) be smooth, compact, oriented manifolds without boundary, both of dimension \(n\). Suppose that \(f:M\to N\) and \(g:M\to N\) are smoothly homotopic. Prove that for any \(\omega \in \Omega ^n(N)\), we have \[ \int _M f^*\omega =\int _M g^*\omega . \]

Raw LaTeX

Let \(M\) and \(N\) be smooth, compact, oriented manifolds without boundary,
both of dimension \(n\). Suppose that \(f:M\to N\) and \(g:M\to N\) are smoothly
homotopic. Prove that for any \(\omega\in\Omega^n(N)\), we have
\[
  \int_M f^*\omega=\int_M g^*\omega.
\]

Topic Tags

Homotopy Equivalence, Retracts, and Contractibility Homotopy and Degree on Manifolds Orientability and Volume Forms Smooth Manifolds and Embedded Submanifolds