Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2009 • Question 10

Question

Math 6510 • Differentiable Manifolds Fall 2009 • Question 10

Let \(M\) and \(N\) be smooth, closed, connected \(n\)-manifolds, and let \(f:M\to N\) be a smooth map. If every point of \(N\) is a regular value, show that the cardinality of the pre-image of a point in \(N\) defines a constant function on \(N\).

Raw LaTeX

Let \(M\) and \(N\) be smooth, closed, connected \(n\)-manifolds, and let
\(f:M\to N\) be a smooth map. If every point of \(N\) is a regular value, show that the
cardinality of the pre-image of a point in \(N\) defines a constant function on \(N\).

Topic Tags

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