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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2003 • Question 6

Question

Math 6510 • Differentiable Manifolds Fall 2003 • Question 6

Give a proof of the Boundary Theorem: If a smooth map \(f:X\to Y\) between two closed oriented \(n\)-manifolds extends to a smooth map \(F:W\to Y\) of a compact oriented \((n+1)\)-manifold \(W\) with \(\partial W=X\) then \(\deg f=0\).

Raw LaTeX

Give a proof of the Boundary Theorem: If a smooth map \(f:X\to Y\) between two
closed oriented \(n\)-manifolds extends to a smooth map \(F:W\to Y\) of a compact oriented
\((n+1)\)-manifold \(W\) with \(\partial W=X\) then \(\deg f=0\).

Topic Tags

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