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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Summer 2017 • Question 7

Question

Math 6510 • Differentiable Manifolds Summer 2017 • Question 7

Let \(\Sigma _g\) be a closed, orientable surface of genus \(g\). Given \(p\ge 1\) distinct points \(\{x_1,\ldots ,x_p\}\subseteq \Sigma _g\), let \(\Sigma _{g,p}=\Sigma _g-\{x_1,\ldots ,x_p\}\). Find the cohomology of \(\Sigma _{g,p}\) with integer coefficients, including its cup product structure.

Raw LaTeX

Let \(\Sigma_g\) be a closed, orientable surface of genus \(g\). Given
\(p\ge1\) distinct points \(\{x_1,\ldots,x_p\}\subseteq\Sigma_g\), let
\(\Sigma_{g,p}=\Sigma_g-\{x_1,\ldots,x_p\}\). Find the cohomology of
\(\Sigma_{g,p}\) with integer coefficients, including its cup product structure.

Topic Tags

Cohomology and Cup Products Homology with Coefficients Orientability and Volume Forms Surface Topology and Surface Groups