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Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2017 • Question 8

Question

Math 6510 • Differentiable Manifolds Fall 2017 • Question 8

Let \(\Sigma _g\) be a closed, orientable surface of genus \(g\ge 1\). Prove that \[ \pi _1(\Sigma _g)= \left \langle a_1,b_1,a_2,b_2,\ldots ,a_g,b_g \,\middle |\, \prod _{i=1}^{g}[a_i,b_i] \right \rangle . \]

Raw LaTeX

Let \(\Sigma_g\) be a closed, orientable surface of genus \(g\ge1\). Prove that
\[
  \pi_1(\Sigma_g)=
  \left\langle
    a_1,b_1,a_2,b_2,\ldots,a_g,b_g
    \,\middle|\,
    \prod_{i=1}^{g}[a_i,b_i]
  \right\rangle.
\]

Topic Tags

Fundamental Groups Orientability and Volume Forms Smooth Manifolds and Embedded Submanifolds Surface Topology and Surface Groups