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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6520 • Algebraic Topology

Summer 2023 • Question 2

Question

Math 6520 • Algebraic Topology Summer 2023 • Question 2

Let \(\Sigma _2\) be the closed, orientable genus two surface. Show that \(\pi _1(\Sigma _2,x)\) has a normal subgroup \(G\) with the quotient group \(\pi _1(\Sigma _2)/G\) isomor- phic to \(\mathbb {Z}\). You can use a picture as part of your proof.

Raw LaTeX

Let \(\Sigma_2\) be the closed, orientable genus two surface. Show that \(\pi_1(\Sigma_2,x)\)
has a normal subgroup \(G\) with the quotient group \(\pi_1(\Sigma_2)/G\) isomor-
phic to \(\mathbb{Z}\). You can use a picture as part of your proof.

Topic Tags

Fundamental Groups Homotopy Groups Surface Topology and Surface Groups