Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2012 • Question 6

Question

Math 6510 • Differentiable Manifolds Spring 2012 • Question 6

Let \(f:M\to N\) be a smooth map between smooth manifolds, and let \(y\) be a regular value of this map. If \(M\) and \(N\) are compact, oriented, and have the same dimension, then what is the definition of \(\deg (f;y)\), the degree of \(f\) measured at the value \(y\)? We have seen that the degree is independent of the regular value chosen, and an ingredient of that proof is the following lemma:

Suppose \(Q\) is a compact smooth manifold and that \(\partial Q=M\). If \(F:Q\to N\) is smooth, and \(z\) is a regular value of \(F|_{\partial Q}\), then \(\deg (F|_{\partial Q};z)=0\).

Prove the lemma.

Let \(f:M\to N\) be a smooth map between smooth manifolds, and let \(y\) be a
regular value of this map. If \(M\) and \(N\) are compact, oriented, and have the
same dimension, then what is the definition of \(\deg(f;y)\), the degree of \(f\)
measured at the value \(y\)? We have seen that the degree is independent of the
regular value chosen, and an ingredient of that proof is the following lemma:

Suppose \(Q\) is a compact smooth manifold and that \(\partial Q=M\). If
\(F:Q\to N\) is smooth, and \(z\) is a regular value of \(F|_{\partial Q}\), then
\(\deg(F|_{\partial Q};z)=0\).

Prove the lemma.

Topic Tags

Homotopy and Degree on Manifolds Orientability and Volume Forms Regular Values and Level Sets Smooth Manifolds and Embedded Submanifolds