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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1999 • Question 1

Question

Math 6510 • Differentiable Manifolds Fall 1999 • Question 1

Let \(f:M\to N\) be a submersion between two smooth manifolds (without boundary).

  1. Show that \(f\) is an open map, i.e., if \(U\subseteq M\) is open, then \(f(U)\subseteq N\) is open.
  2. Show that if \(M\) is closed (i.e. compact without boundary) and \(N\) is connected, then \(f\) is onto.
  3. Show that there is no submersion of a closed manifold to \(\mathbb {R}^n\).

Raw LaTeX

Let \(f:M\to N\) be a submersion between two smooth manifolds (without boundary).
\begin{enumerate}
\item Show that \(f\) is an open map, i.e., if \(U\subseteq M\) is open, then
\(f(U)\subseteq N\) is open.
\item Show that if \(M\) is closed (i.e. compact without boundary) and \(N\) is connected,
then \(f\) is onto.
\item Show that there is no submersion of a closed manifold to \(\mathbb{R}^n\).
\end{enumerate}

Topic Tags

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