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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2002 • Question 2

Question

Math 6510 • Differentiable Manifolds Fall 2002 • Question 2

Consider a compact differentiable \(m\)-manifold \(M\) without boundary and let \(N=M\times M\). Let \(\Delta \subset N\) be the diagonal. Prove that there is no compact differentiable \(m+1\)-dimensional submanifold with boundary \(B\subset N\), so that the boundary of \(B\) is \(\Delta \).

Raw LaTeX

Consider a compact differentiable \(m\)-manifold \(M\) without boundary and let
\(N=M\times M\). Let \(\Delta\subset N\) be the diagonal. Prove that there is no compact
differentiable \(m+1\)-dimensional submanifold with boundary \(B\subset N\), so that the
boundary of \(B\) is \(\Delta\).

Topic Tags

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