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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Summer 2018 • Question 6

Question

Math 6510 • Differentiable Manifolds Summer 2018 • Question 6

Let \(f:M\to N\) and \(g:N\to P\) be diffeomorphisms of smooth manifolds. Let \(\sigma (f)\) equal 1 if \(f\) preserves orientation and \(-1\) otherwise. Defining \(\sigma (g)\) and \(\sigma (g\circ f)\) similarly, prove that \(\sigma (g\circ f)=\sigma (g)\sigma (f)\).

Raw LaTeX

Let \(f:M\to N\) and \(g:N\to P\) be diffeomorphisms of smooth manifolds. Let
\(\sigma(f)\) equal 1 if \(f\) preserves orientation and \(-1\) otherwise. Defining
\(\sigma(g)\) and \(\sigma(g\circ f)\) similarly, prove that
\(\sigma(g\circ f)=\sigma(g)\sigma(f)\).

Topic Tags

Diffeomorphisms and Isotopy Methods Homotopy and Degree on Manifolds Orientability and Volume Forms Smooth Manifolds and Embedded Submanifolds