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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2013 • Question 11

Question

Math 6510 • Differentiable Manifolds Spring 2013 • Question 11

Let \(S_g\) be the closed connected oriented surface of genus \(g\).

  1. Prove that for \(g<h\) every map \(S_g\to S_h\) has degree 0.
  2. Prove or disprove: for \(g<h\) every map \(S_g\to S_h\) is nullhomotopic?

Raw LaTeX

Let \(S_g\) be the closed connected oriented surface of genus \(g\).
\begin{enumerate}
\item[(a)] Prove that for \(g<h\) every map \(S_g\to S_h\) has degree 0.
\item[(b)] Prove or disprove: for \(g<h\) every map \(S_g\to S_h\) is nullhomotopic?
\end{enumerate}

Topic Tags

Homotopy Equivalence, Retracts, and Contractibility Orientability and Volume Forms Smooth Manifolds and Embedded Submanifolds Surface Topology and Surface Groups