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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2018 • Question 1

Question

Math 6510 • Differentiable Manifolds Spring 2018 • Question 1

Let \(M\) and \(N\) be smooth manifolds. If \(f:M\to N\) is continuous, proper, and injective, prove that \(f^{-1}:f(M)\to M\) is continuous. Conclude that any proper, injective, immersion \(f:M\to N\) is an embedding.

Raw LaTeX

Let \(M\) and \(N\) be smooth manifolds. If \(f:M\to N\) is continuous, proper,
and injective, prove that \(f^{-1}:f(M)\to M\) is continuous. Conclude that any
proper, injective, immersion \(f:M\to N\) is an embedding.

Topic Tags

Homotopy and Degree on Manifolds Immersions, Submersions, and Embeddings Smooth Manifolds and Embedded Submanifolds Stokes' Theorem and Integration on Manifolds