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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1995 • Question 12

Question

Math 6510 • Differentiable Manifolds Fall 1995 • Question 12

Let \(f:M\to N\) and \(g:N\to M\) be smooth maps, where \(M\) and \(N\) are compact connected smooth oriented manifolds of the same dimension. If \(g\circ f\) is a diffeomorphism, either prove that both \(f\) and \(g\) are diffeomorphisms, or find a counterexample.

Raw LaTeX

Let \(f:M\to N\) and \(g:N\to M\) be smooth maps, where \(M\) and \(N\) are
compact connected smooth oriented manifolds of the same dimension. If \(g\circ f\) is
a diffeomorphism, either prove that both \(f\) and \(g\) are diffeomorphisms, or find a
counterexample.

Topic Tags

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