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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2000 • Question 9

Question

Math 6510 • Differentiable Manifolds Fall 2000 • Question 9

A. Let \(F_g\) be a closed orientable surface of genus \(g\). Prove that \(\pi _1(F_g)\) is infinite, and non-abelian, provided \(g>1\).

B. Let \(G\) be the one point union of two circles (a figure eight). Draw a picture of an explicit irregular covering map \(\widetilde {G}\to G\).

Raw LaTeX

\textbf{A.} Let \(F_g\) be a closed orientable surface of genus \(g\). Prove that
\(\pi_1(F_g)\) is infinite, and non-abelian, provided \(g>1\).

\textbf{B.} Let \(G\) be the one point union of two circles (a figure eight). Draw a
picture of an explicit irregular covering map \(\widetilde{G}\to G\).

Topic Tags

Covering Spaces Orientability and Volume Forms Smooth Manifolds and Embedded Submanifolds Surface Topology and Surface Groups