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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Winter 2004 • Question 14

Question

Math 6510 • Differentiable Manifolds Winter 2004 • Question 14

Let \(S\) be a closed, orientable surface and \(\gamma \) a simple closed curve on \(S\). Show that \(S\) retracts onto \(\gamma \) if and only if \(\gamma \) represents a non-trivial element in homology.

Raw LaTeX

Let \(S\) be a closed, orientable surface and \(\gamma\) a simple closed curve on
\(S\). Show that \(S\) retracts onto \(\gamma\) if and only if \(\gamma\) represents a
non-trivial element in homology.

Topic Tags

Homology with Coefficients Homotopy Equivalence, Retracts, and Contractibility Orientability and Volume Forms Surface Topology and Surface Groups