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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2011 • Question 5

Question

Math 6510 • Differentiable Manifolds Spring 2011 • Question 5

Let \(f:S^1\to S^1\) be given by \(f(z)=z^2\), where we regard \(S^1\) as the unit circle in \(\mathbb {C}\). Find a presentation of the fundamental group of the mapping torus \[ M_f=S^1\times I/(z,1)\sim (f(z),0) \] of \(f\).

Raw LaTeX

Let \(f:S^1\to S^1\) be given by \(f(z)=z^2\), where we regard \(S^1\) as the
unit circle in \(\C\). Find a presentation of the fundamental group of the mapping
torus
\[
  M_f=S^1\times I/(z,1)\sim(f(z),0)
\]
of \(f\).

Topic Tags

Free Groups, Graph Coverings, and Presentations Fundamental Groups Mapping Tori Surface Topology and Surface Groups