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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Winter 2004 • Question 9

Question

Math 6510 • Differentiable Manifolds Winter 2004 • Question 9

Let \(f:\Sigma \to S^1\) be a smooth map from the surface of genus 2 to the circle. Suppose that \(y\in S^1\) is a regular value of \(f\) and that \(f^{-1}(y)\) is a nonseparating circle in \(\Sigma \) (i.e. \(\Sigma \setminus f^{-1}(y)\) is connected). Prove that \(f\) is not nullhomotopic.

Raw LaTeX

Let \(f:\Sigma\to S^1\) be a smooth map from the surface of genus 2 to the circle.
Suppose that \(y\in S^1\) is a regular value of \(f\) and that \(f^{-1}(y)\) is a
nonseparating circle in \(\Sigma\) (i.e. \(\Sigma\setminus f^{-1}(y)\) is connected). Prove
that \(f\) is not nullhomotopic.

Topic Tags

Homotopy Equivalence, Retracts, and Contractibility Regular Values and Level Sets Smooth Manifolds and Embedded Submanifolds Surface Topology and Surface Groups