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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Winter 2004 • Question 13

Question

Math 6510 • Differentiable Manifolds Winter 2004 • Question 13

  1. Let \(M\) be a manifold with boundary. Show that if \(M\) is orientable then \(\partial M\) is orientable.
  2. Let \(M\) be an \(n\)-dimensional manifold and \(N\subset M\) an \(n-1\) dimensional submanifold such that \(N\) has a neighborhood \(U\subset M\) with \(U\) homeomorphic to \(N\times (0,1)\). Show that if \(M\) is orientable then \(N\) is orientable.

Raw LaTeX

\begin{enumerate}
\item Let \(M\) be a manifold with boundary. Show that if \(M\) is orientable then
\(\partial M\) is orientable.
\item Let \(M\) be an \(n\)-dimensional manifold and \(N\subset M\) an \(n-1\) dimensional
submanifold such that \(N\) has a neighborhood \(U\subset M\) with \(U\) homeomorphic to
\(N\times(0,1)\). Show that if \(M\) is orientable then \(N\) is orientable.
\end{enumerate}

Topic Tags

Homotopy and Degree on Manifolds Orientability and Volume Forms Smooth Manifolds and Embedded Submanifolds Stokes' Theorem and Integration on Manifolds