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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2016 • Question 7

Question

Math 6510 • Differentiable Manifolds Spring 2016 • Question 7

Show that \(\pi _1(S^1)=\mathbb {Z}\); that \(\pi _1(S^n)=1\) if \(n\ge 2\); that \(\pi _1(\Sigma _1-\{x\})=F_2\) (where \(\Sigma _1-\{x\}\) is a closed orientable genus 1 surface with a single point removed and \(F_2\) is a free group on two generators); and find \(\pi _1(\Sigma _2)\) (where \(\Sigma _2\) is a closed orientable surface of genus 2).

Raw LaTeX

Show that \(\pi_1(S^1)=\Z\); that \(\pi_1(S^n)=1\) if \(n\ge2\); that
\(\pi_1(\Sigma_1-\{x\})=F_2\) (where \(\Sigma_1-\{x\}\) is a closed orientable genus
1 surface with a single point removed and \(F_2\) is a free group on two
generators); and find \(\pi_1(\Sigma_2)\) (where \(\Sigma_2\) is a closed orientable
surface of genus 2).

Topic Tags

Free Groups, Graph Coverings, and Presentations Orientability and Volume Forms Smooth Manifolds and Embedded Submanifolds Surface Topology and Surface Groups