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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2015 • Question 4

Question

Math 6510 • Differentiable Manifolds Spring 2015 • Question 4

Let \(f:M\to N\) be a differentiable map between \(n\)-manifolds and assume that \(M\) is compact and \(N\) is connected. If \(x\in N\) is a regular value of \(f\) show that \(f^{-1}(x)\) is finitely many points in \(M\). If every point in \(N\) is a regular value of \(f\) define a function \(\#:N\to \mathbb {Z}\) by letting \(\#(x)\) be the number of points in \(f^{-1}(x)\) and show that \(\#\) is constant.

Raw LaTeX

Let \(f:M\to N\) be a differentiable map between \(n\)-manifolds and assume
that \(M\) is compact and \(N\) is connected. If \(x\in N\) is a regular value of \(f\)
show that \(f^{-1}(x)\) is finitely many points in \(M\). If every point in \(N\) is a
regular value of \(f\) define a function \(\#:N\to\Z\) by letting \(\#(x)\) be the
number of points in \(f^{-1}(x)\) and show that \(\#\) is constant.

Topic Tags

Homotopy and Degree on Manifolds Regular Values and Level Sets Smooth Manifolds and Embedded Submanifolds Stokes' Theorem and Integration on Manifolds