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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Summer 2016 • Question 11

Question

Math 6510 • Differentiable Manifolds Summer 2016 • Question 11

Describe the cohomology ring \(H^*(\Sigma _g,\mathbb {Z})\) (i.e. the groups together with the cup product) of a compact oriented surface \(\Sigma _g\) of genus \(g\).

  1. Prove that if \(g<h\), then every map \(f:\Sigma _g\to \Sigma _h\) has degree zero.
  2. Show that for any \(g\ge h\), there is a map \(f:\Sigma _g\to \Sigma _h\) of degree one.

Raw LaTeX

Describe the cohomology ring \(H^*(\Sigma_g,\Z)\) (i.e. the groups together
with the cup product) of a compact oriented surface \(\Sigma_g\) of genus \(g\).
\begin{enumerate}
\item[(a)] Prove that if \(g<h\), then every map \(f:\Sigma_g\to\Sigma_h\) has
degree zero.
\item[(b)] Show that for any \(g\ge h\), there is a map \(f:\Sigma_g\to\Sigma_h\)
of degree one.
\end{enumerate}

Topic Tags

Cohomology and Cup Products Orientability and Volume Forms Smooth Manifolds and Embedded Submanifolds Surface Topology and Surface Groups