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Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2009 • Question 17

Question

Math 6510 • Differentiable Manifolds Fall 2009 • Question 17

Compute the fundamental group of the space \(X_{2,3}\) obtained from the cylinder \(S^1\times [-1,1]\) via identifications: \((z,-1)\sim (-z,-1)\) and \[ (z,1)\sim \left (ze^{\frac {2\pi i}{3}},1\right ) \sim \left (ze^{\frac {4\pi i}{3}},1\right ) \] for all \(z\in S^1=\{z\in \mathbb {C}\mid |z|=1\}\). (That is, in the circle \(S^1\times \{-1\}\) antipodal points are identified, and in \(S^1\times \{1\}\) two points are identified if they differ by an order 3 rotation.) Show that this group is infinite.

Raw LaTeX

Compute the fundamental group of the space \(X_{2,3}\) obtained from the
cylinder \(S^1\times[-1,1]\) via identifications:
\((z,-1)\sim(-z,-1)\) and
\[
  (z,1)\sim\left(ze^{\frac{2\pi i}{3}},1\right)
  \sim\left(ze^{\frac{4\pi i}{3}},1\right)
\]
for all \(z\in S^1=\{z\in\C\mid |z|=1\}\). (That is, in the circle
\(S^1\times\{-1\}\) antipodal points are identified, and in \(S^1\times\{1\}\) two
points are identified if they differ by an order 3 rotation.) Show that this group
is infinite.

Topic Tags

Fundamental Groups